Eigenvalues and eigenvectors

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id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenvalues_and_eigenvectors_of_matrices" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Eigenvalues_and_eigenvectors_of_matrices"> <div class="vector-toc-text"> 4 Eigenvalues and eigenvectors of matrices </div> </a> <button aria-controls="toc-Eigenvalues_and_eigenvectors_of_matrices-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Eigenvalues and eigenvectors of matrices subsection </button> <ul id="toc-Eigenvalues_and_eigenvectors_of_matrices-sublist" class="vector-toc-list"> <li id="toc-Eigenvalues_and_the_characteristic_polynomial" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenvalues_and_the_characteristic_polynomial"> <div class="vector-toc-text"> 4.1 Eigenvalues and the characteristic polynomial </div> </a> <ul id="toc-Eigenvalues_and_the_characteristic_polynomial-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spectrum_of_a_matrix" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spectrum_of_a_matrix"> <div class="vector-toc-text"> 4.2 Spectrum of a matrix </div> </a> <ul id="toc-Spectrum_of_a_matrix-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Algebraic_multiplicity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Algebraic_multiplicity"> <div class="vector-toc-text"> 4.3 Algebraic multiplicity </div> </a> <ul id="toc-Algebraic_multiplicity-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenspaces,_geometric_multiplicity,_and_the_eigenbasis_for_matrices" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenspaces,_geometric_multiplicity,_and_the_eigenbasis_for_matrices"> <div class="vector-toc-text"> 4.4 Eigenspaces, geometric multiplicity, and the eigenbasis for matrices </div> </a> <ul id="toc-Eigenspaces,_geometric_multiplicity,_and_the_eigenbasis_for_matrices-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_properties" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Additional_properties"> <div class="vector-toc-text"> 4.5 Additional properties </div> </a> <ul id="toc-Additional_properties-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Left_and_right_eigenvectors" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Left_and_right_eigenvectors"> <div class="vector-toc-text"> 4.6 Left and right eigenvectors </div> </a> <ul id="toc-Left_and_right_eigenvectors-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diagonalization_and_the_eigendecomposition" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Diagonalization_and_the_eigendecomposition"> <div class="vector-toc-text"> 4.7 Diagonalization and the eigendecomposition </div> </a> <ul id="toc-Diagonalization_and_the_eigendecomposition-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Variational_characterization" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Variational_characterization"> <div class="vector-toc-text"> 4.8 Variational characterization </div> </a> <ul id="toc-Variational_characterization-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_examples" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Matrix_examples"> <div class="vector-toc-text"> 4.9 Matrix examples </div> </a> <ul id="toc-Matrix_examples-sublist" class="vector-toc-list"> <li id="toc-Two-dimensional_matrix_example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Two-dimensional_matrix_example"> <div class="vector-toc-text"> 4.9.1 Two-dimensional matrix example </div> </a> <ul id="toc-Two-dimensional_matrix_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Three-dimensional_matrix_example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Three-dimensional_matrix_example"> <div class="vector-toc-text"> 4.9.2 Three-dimensional matrix example </div> </a> <ul id="toc-Three-dimensional_matrix_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Three-dimensional_matrix_example_with_complex_eigenvalues" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Three-dimensional_matrix_example_with_complex_eigenvalues"> <div class="vector-toc-text"> 4.9.3 Three-dimensional matrix example with complex eigenvalues </div> </a> <ul id="toc-Three-dimensional_matrix_example_with_complex_eigenvalues-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Diagonal_matrix_example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Diagonal_matrix_example"> <div class="vector-toc-text"> 4.9.4 Diagonal matrix example </div> </a> <ul id="toc-Diagonal_matrix_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Triangular_matrix_example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Triangular_matrix_example"> <div class="vector-toc-text"> 4.9.5 Triangular matrix example </div> </a> <ul id="toc-Triangular_matrix_example-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Matrix_with_repeated_eigenvalues_example" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Matrix_with_repeated_eigenvalues_example"> <div class="vector-toc-text"> 4.9.6 Matrix with repeated eigenvalues example </div> </a> <ul id="toc-Matrix_with_repeated_eigenvalues_example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Eigenvector-eigenvalue_identity" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenvector-eigenvalue_identity"> <div class="vector-toc-text"> 4.10 Eigenvector-eigenvalue identity </div> </a> <ul id="toc-Eigenvector-eigenvalue_identity-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Eigenvalues_and_eigenfunctions_of_differential_operators" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Eigenvalues_and_eigenfunctions_of_differential_operators"> <div class="vector-toc-text"> 5 Eigenvalues and eigenfunctions of differential operators </div> </a> <button aria-controls="toc-Eigenvalues_and_eigenfunctions_of_differential_operators-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Eigenvalues and eigenfunctions of differential operators subsection </button> <ul id="toc-Eigenvalues_and_eigenfunctions_of_differential_operators-sublist" class="vector-toc-list"> <li id="toc-Derivative_operator_example" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Derivative_operator_example"> <div class="vector-toc-text"> 5.1 Derivative operator example </div> </a> <ul id="toc-Derivative_operator_example-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-General_definition" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#General_definition"> <div class="vector-toc-text"> 6 General definition </div> </a> <button aria-controls="toc-General_definition-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle General definition subsection </button> <ul id="toc-General_definition-sublist" class="vector-toc-list"> <li id="toc-Eigenspaces,_geometric_multiplicity,_and_the_eigenbasis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenspaces,_geometric_multiplicity,_and_the_eigenbasis"> <div class="vector-toc-text"> 6.1 Eigenspaces, geometric multiplicity, and the eigenbasis </div> </a> <ul id="toc-Eigenspaces,_geometric_multiplicity,_and_the_eigenbasis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Spectral_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Spectral_theory"> <div class="vector-toc-text"> 6.2 Spectral theory </div> </a> <ul id="toc-Spectral_theory-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Associative_algebras_and_representation_theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Associative_algebras_and_representation_theory"> <div class="vector-toc-text"> 6.3 Associative algebras and representation theory </div> </a> <ul id="toc-Associative_algebras_and_representation_theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Dynamic_equations" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Dynamic_equations"> <div class="vector-toc-text"> 7 Dynamic equations </div> </a> <ul id="toc-Dynamic_equations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Calculation" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Calculation"> <div class="vector-toc-text"> 8 Calculation </div> </a> <button aria-controls="toc-Calculation-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Calculation subsection </button> <ul id="toc-Calculation-sublist" class="vector-toc-list"> <li id="toc-Classical_method" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Classical_method"> <div class="vector-toc-text"> 8.1 Classical method </div> </a> <ul id="toc-Classical_method-sublist" class="vector-toc-list"> <li id="toc-Eigenvalues" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Eigenvalues"> <div class="vector-toc-text"> 8.1.1 Eigenvalues </div> </a> <ul id="toc-Eigenvalues-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenvectors" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Eigenvectors"> <div class="vector-toc-text"> 8.1.2 Eigenvectors </div> </a> <ul id="toc-Eigenvectors-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Simple_iterative_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Simple_iterative_methods"> <div class="vector-toc-text"> 8.2 Simple iterative methods </div> </a> <ul id="toc-Simple_iterative_methods-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_methods" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_methods"> <div class="vector-toc-text"> 8.3 Modern methods </div> </a> <ul id="toc-Modern_methods-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Applications" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Applications"> <div class="vector-toc-text"> 9 Applications </div> </a> <button aria-controls="toc-Applications-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Applications subsection </button> <ul id="toc-Applications-sublist" class="vector-toc-list"> <li id="toc-Geometric_transformations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geometric_transformations"> <div class="vector-toc-text"> 9.1 Geometric transformations </div> </a> <ul id="toc-Geometric_transformations-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Principal_component_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Principal_component_analysis"> <div class="vector-toc-text"> 9.2 Principal component analysis </div> </a> <ul id="toc-Principal_component_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Graphs" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Graphs"> <div class="vector-toc-text"> 9.3 Graphs </div> </a> <ul id="toc-Graphs-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Markov_chains" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Markov_chains"> <div class="vector-toc-text"> 9.4 Markov chains </div> </a> <ul id="toc-Markov_chains-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Vibration_analysis" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Vibration_analysis"> <div class="vector-toc-text"> 9.5 Vibration analysis </div> </a> <ul id="toc-Vibration_analysis-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Tensor_of_moment_of_inertia" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Tensor_of_moment_of_inertia"> <div class="vector-toc-text"> 9.6 Tensor of moment of inertia </div> </a> <ul id="toc-Tensor_of_moment_of_inertia-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Stress_tensor" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stress_tensor"> <div class="vector-toc-text"> 9.7 Stress tensor </div> </a> <ul id="toc-Stress_tensor-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Schrödinger_equation" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Schrödinger_equation"> <div class="vector-toc-text"> 9.8 Schrödinger equation </div> </a> <ul id="toc-Schrödinger_equation-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Wave_transport" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Wave_transport"> <div class="vector-toc-text"> 9.9 Wave transport </div> </a> <ul id="toc-Wave_transport-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Molecular_orbitals" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Molecular_orbitals"> <div class="vector-toc-text"> 9.10 Molecular orbitals </div> </a> <ul id="toc-Molecular_orbitals-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Geology_and_glaciology" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Geology_and_glaciology"> <div class="vector-toc-text"> 9.11 Geology and glaciology </div> </a> <ul id="toc-Geology_and_glaciology-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Basic_reproduction_number" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Basic_reproduction_number"> <div class="vector-toc-text"> 9.12 Basic reproduction number </div> </a> <ul id="toc-Basic_reproduction_number-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Eigenfaces" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Eigenfaces"> <div class="vector-toc-text"> 9.13 Eigenfaces </div> </a> <ul id="toc-Eigenfaces-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> 10 See also </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Notes" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Notes"> <div class="vector-toc-text"> 11 Notes </div> </a> <button aria-controls="toc-Notes-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Notes subsection </button> <ul id="toc-Notes-sublist" class="vector-toc-list"> <li id="toc-Citations" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Citations"> <div class="vector-toc-text"> 11.1 Citations </div> </a> <ul id="toc-Citations-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Sources" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Sources"> <div class="vector-toc-text"> 12 Sources </div> </a> <ul id="toc-Sources-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Further_reading" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#Further_reading"> <div class="vector-toc-text"> 13 Further reading </div> </a> <ul id="toc-Further_reading-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 14 External links </div> </a> <button aria-controls="toc-External_links-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle External links subsection </button> <ul id="toc-External_links-sublist" class="vector-toc-list"> <li id="toc-Theory" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Theory"> <div class="vector-toc-text"> 14.1 Theory </div> </a> <ul id="toc-Theory-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </div> </div> </nav> </div> </div> <div class="mw-content-container"> <main id="content" class="mw-body"> <header 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propiu y valor propiu – Asturian" lang="ast" hreflang="ast" data-title="Vector propiu y valor propiu" data-language-autonym="Asturianu" data-language-local-name="Asturian" class="interlanguage-link-target">Asturianu</a></li><li class="interlanguage-link interwiki-be mw-list-item"><a href="https://be.wikipedia.org/wiki/%D0%A3%D0%BB%D0%B0%D1%81%D0%BD%D1%8B%D1%8F_%D0%B2%D0%B5%D0%BA%D1%82%D0%B0%D1%80%D1%8B_%D1%96_%D1%9E%D0%BB%D0%B0%D1%81%D0%BD%D1%8B%D1%8F_%D0%B7%D0%BD%D0%B0%D1%87%D1%8D%D0%BD%D0%BD%D1%96" title="Уласныя вектары і ўласныя значэнні – Belarusian" lang="be" hreflang="be" data-title="Уласныя вектары і ўласныя значэнні" data-language-autonym="Беларуская" data-language-local-name="Belarusian" class="interlanguage-link-target">Беларуская</a></li><li class="interlanguage-link interwiki-be-x-old mw-list-item"><a href="https://be-tarask.wikipedia.org/wiki/%D0%A3%D0%BB%D0%B0%D1%81%D0%BD%D1%8B%D1%8F_%D0%BB%D1%96%D0%BA%D1%96,_%D0%B2%D1%8D%D0%BA%D1%82%D0%B0%D1%80%D1%8B_%D1%96_%D0%BF%D1%80%D0%B0%D1%81%D1%82%D0%BE%D1%80%D1%8B" title="Уласныя лікі, вэктары і прасторы – Belarusian (Taraškievica orthography)" lang="be-tarask" hreflang="be-tarask" data-title="Уласныя лікі, вэктары і прасторы" data-language-autonym="Беларуская (тарашкевіца)" data-language-local-name="Belarusian (Taraškievica orthography)" class="interlanguage-link-target">Беларуская (тарашкевіца)</a></li><li class="interlanguage-link interwiki-bg mw-list-item"><a href="https://bg.wikipedia.org/wiki/%D0%A1%D0%BE%D0%B1%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%B8_%D1%81%D1%82%D0%BE%D0%B9%D0%BD%D0%BE%D1%81%D1%82%D0%B8_%D0%B8_%D1%81%D0%BE%D0%B1%D1%81%D1%82%D0%B2%D0%B5%D0%BD%D0%B8_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8" title="Собствени стойности и собствени вектори – Bulgarian" lang="bg" hreflang="bg" data-title="Собствени стойности и собствени вектори" data-language-autonym="Български" data-language-local-name="Bulgarian" class="interlanguage-link-target">Български</a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Valor_propi,_vector_propi_i_espai_propi" title="Valor propi, vector propi i espai propi – Catalan" lang="ca" hreflang="ca" data-title="Valor propi, vector propi i espai propi" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target">Català</a></li><li class="interlanguage-link interwiki-cs mw-list-item"><a href="https://cs.wikipedia.org/wiki/Vlastn%C3%AD_vektory_a_vlastn%C3%AD_%C4%8D%C3%ADsla" title="Vlastní vektory a vlastní čísla – Czech" lang="cs" hreflang="cs" data-title="Vlastní vektory a vlastní čísla" data-language-autonym="Čeština" data-language-local-name="Czech" class="interlanguage-link-target">Čeština</a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Egenv%C3%A6rdi,_egenvektor_og_egenrum" title="Egenværdi, egenvektor og egenrum – Danish" lang="da" hreflang="da" data-title="Egenværdi, egenvektor og egenrum" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target">Dansk</a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Eigenwerte_und_Eigenvektoren" title="Eigenwerte und Eigenvektoren – German" lang="de" hreflang="de" data-title="Eigenwerte und Eigenvektoren" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target">Deutsch</a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/Omav%C3%A4%C3%A4rtus_ja_omavektor" title="Omaväärtus ja omavektor – Estonian" lang="et" hreflang="et" data-title="Omaväärtus ja omavektor" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target">Eesti</a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%99%CE%B4%CE%B9%CE%BF%CF%84%CE%B9%CE%BC%CE%AD%CF%82_%CE%BA%CE%B1%CE%B9_%CE%B9%CE%B4%CE%B9%CE%BF%CE%B4%CE%B9%CE%B1%CE%BD%CF%8D%CF%83%CE%BC%CE%B1%CF%84%CE%B1" title="Ιδιοτιμές και ιδιοδιανύσματα – Greek" lang="el" hreflang="el" data-title="Ιδιοτιμές και ιδιοδιανύσματα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target">Ελληνικά</a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Vector,_valor_y_espacio_propios" title="Vector, valor y espacio propios – Spanish" lang="es" hreflang="es" data-title="Vector, valor y espacio propios" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target">Español</a></li><li class="interlanguage-link interwiki-eo mw-list-item"><a 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href="https://ko.wikipedia.org/wiki/%EA%B3%A0%EC%9C%B3%EA%B0%92%EA%B3%BC_%EA%B3%A0%EC%9C%A0_%EB%B2%A1%ED%84%B0" title="고윳값과 고유 벡터 – Korean" lang="ko" hreflang="ko" data-title="고윳값과 고유 벡터" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target">한국어</a></li><li class="interlanguage-link interwiki-hi mw-list-item"><a href="https://hi.wikipedia.org/wiki/%E0%A4%85%E0%A4%AD%E0%A4%BF%E0%A4%B2%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%A3%E0%A4%BF%E0%A4%95_%E0%A4%AE%E0%A4%BE%E0%A4%A8_%E0%A4%A4%E0%A4%A5%E0%A4%BE_%E0%A4%85%E0%A4%AD%E0%A4%BF%E0%A4%B2%E0%A4%95%E0%A5%8D%E0%A4%B7%E0%A4%A3%E0%A4%BF%E0%A4%95_%E0%A4%B8%E0%A4%A6%E0%A4%BF%E0%A4%B6" title="अभिलक्षणिक मान तथा अभिलक्षणिक सदिश – Hindi" lang="hi" hreflang="hi" data-title="अभिलक्षणिक मान तथा अभिलक्षणिक सदिश" data-language-autonym="हिन्दी" data-language-local-name="Hindi" class="interlanguage-link-target">हिन्दी</a></li><li class="interlanguage-link interwiki-hr mw-list-item"><a href="https://hr.wikipedia.org/wiki/Svojstvene_vrijednosti_i_svojstveni_vektori" title="Svojstvene vrijednosti i svojstveni vektori – Croatian" lang="hr" hreflang="hr" data-title="Svojstvene vrijednosti i svojstveni vektori" data-language-autonym="Hrvatski" data-language-local-name="Croatian" class="interlanguage-link-target">Hrvatski</a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Nilai_dan_vektor_eigen" title="Nilai dan vektor eigen – Indonesian" lang="id" hreflang="id" data-title="Nilai dan vektor eigen" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target">Bahasa Indonesia</a></li><li class="interlanguage-link interwiki-ia mw-list-item"><a href="https://ia.wikipedia.org/wiki/Eigenvalores_e_eigenvectores" title="Eigenvalores e eigenvectores – Interlingua" lang="ia" hreflang="ia" data-title="Eigenvalores e eigenvectores" data-language-autonym="Interlingua" data-language-local-name="Interlingua" class="interlanguage-link-target">Interlingua</a></li><li class="interlanguage-link interwiki-is mw-list-item"><a href="https://is.wikipedia.org/wiki/Eigen_gildi" title="Eigen gildi – Icelandic" lang="is" hreflang="is" data-title="Eigen gildi" data-language-autonym="Íslenska" data-language-local-name="Icelandic" class="interlanguage-link-target">Íslenska</a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Autovettore_e_autovalore" title="Autovettore e autovalore – Italian" lang="it" hreflang="it" data-title="Autovettore e autovalore" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target">Italiano</a></li><li class="interlanguage-link interwiki-lv mw-list-item"><a href="https://lv.wikipedia.org/wiki/%C4%AApa%C5%A1v%C4%93rt%C4%ABbas_un_%C4%ABpa%C5%A1vektori" title="Īpašvērtības un īpašvektori – Latvian" lang="lv" hreflang="lv" data-title="Īpašvērtības un īpašvektori" data-language-autonym="Latviešu" data-language-local-name="Latvian" class="interlanguage-link-target">Latviešu</a></li><li class="interlanguage-link interwiki-lt mw-list-item"><a href="https://lt.wikipedia.org/wiki/Tikrini%C5%B3_ver%C4%8Di%C5%B3_lygtis" title="Tikrinių verčių lygtis – Lithuanian" lang="lt" hreflang="lt" data-title="Tikrinių verčių lygtis" data-language-autonym="Lietuvių" data-language-local-name="Lithuanian" class="interlanguage-link-target">Lietuvių</a></li><li class="interlanguage-link interwiki-hu mw-list-item"><a href="https://hu.wikipedia.org/wiki/Saj%C3%A1tvektor_%C3%A9s_saj%C3%A1t%C3%A9rt%C3%A9k" title="Sajátvektor és sajátérték – Hungarian" lang="hu" hreflang="hu" data-title="Sajátvektor és sajátérték" data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target">Magyar</a></li><li class="interlanguage-link interwiki-ja mw-list-item"><a href="https://ja.wikipedia.org/wiki/%E5%9B%BA%E6%9C%89%E5%80%A4%E3%81%A8%E5%9B%BA%E6%9C%89%E3%83%99%E3%82%AF%E3%83%88%E3%83%AB" title="固有値と固有ベクトル – Japanese" lang="ja" hreflang="ja" data-title="固有値と固有ベクトル" data-language-autonym="日本語" data-language-local-name="Japanese" class="interlanguage-link-target">日本語</a></li><li class="interlanguage-link interwiki-nn mw-list-item"><a href="https://nn.wikipedia.org/wiki/Eigenverdi,_eigenvektor_og_eigerom" title="Eigenverdi, eigenvektor og eigerom – Norwegian Nynorsk" lang="nn" hreflang="nn" data-title="Eigenverdi, eigenvektor og eigerom" data-language-autonym="Norsk nynorsk" data-language-local-name="Norwegian Nynorsk" class="interlanguage-link-target">Norsk nynorsk</a></li><li class="interlanguage-link interwiki-pa mw-list-item"><a href="https://pa.wikipedia.org/wiki/%E0%A8%86%E0%A8%88%E0%A8%97%E0%A8%A8-%E0%A8%AE%E0%A9%81%E0%A9%B1%E0%A8%B2_%E0%A8%85%E0%A8%A4%E0%A9%87_%E0%A8%86%E0%A8%88%E0%A8%97%E0%A8%A8-%E0%A8%B5%E0%A9%88%E0%A8%95%E0%A8%9F%E0%A8%B0" title="ਆਈਗਨ-ਮੁੱਲ ਅਤੇ ਆਈਗਨ-ਵੈਕਟਰ – Punjabi" lang="pa" hreflang="pa" data-title="ਆਈਗਨ-ਮੁੱਲ ਅਤੇ ਆਈਗਨ-ਵੈਕਟਰ" data-language-autonym="ਪੰਜਾਬੀ" data-language-local-name="Punjabi" class="interlanguage-link-target">ਪੰਜਾਬੀ</a></li><li class="interlanguage-link interwiki-pnb mw-list-item"><a href="https://pnb.wikipedia.org/wiki/%D9%88%DB%8C%DA%98%DB%81_%D9%82%D8%AF%D8%B1" title="ویژہ قدر – Western Punjabi" lang="pnb" hreflang="pnb" data-title="ویژہ قدر" data-language-autonym="پنجابی" data-language-local-name="Western Punjabi" class="interlanguage-link-target">پنجابی</a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Wektory_i_warto%C5%9Bci_w%C5%82asne" title="Wektory i wartości własne – Polish" lang="pl" hreflang="pl" data-title="Wektory i wartości własne" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target">Polski</a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Autovalores_e_autovetores" title="Autovalores e autovetores – Portuguese" lang="pt" hreflang="pt" data-title="Autovalores e autovetores" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target">Português</a></li><li class="interlanguage-link interwiki-ro mw-list-item"><a href="https://ro.wikipedia.org/wiki/Vectori_%C8%99i_valori_proprii" title="Vectori și valori proprii – Romanian" lang="ro" hreflang="ro" data-title="Vectori și valori proprii" data-language-autonym="Română" data-language-local-name="Romanian" class="interlanguage-link-target">Română</a></li><li class="interlanguage-link interwiki-sq mw-list-item"><a href="https://sq.wikipedia.org/wiki/Autovlerat_dhe_autovektor%C3%ABt" title="Autovlerat dhe autovektorët – Albanian" lang="sq" hreflang="sq" data-title="Autovlerat dhe autovektorët" data-language-autonym="Shqip" data-language-local-name="Albanian" class="interlanguage-link-target">Shqip</a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors – Simple English" lang="en-simple" hreflang="en-simple" data-title="Eigenvalues and eigenvectors" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target">Simple English</a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Lastna_vrednost" title="Lastna vrednost – Slovenian" lang="sl" hreflang="sl" data-title="Lastna vrednost" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target">Slovenščina</a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/Sopstvene_vrednosti_i_sopstveni_vektori" title="Sopstvene vrednosti i sopstveni vektori – Serbian" lang="sr" hreflang="sr" data-title="Sopstvene vrednosti i sopstveni vektori" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target">Српски / srpski</a></li><li class="interlanguage-link interwiki-sh mw-list-item"><a href="https://sh.wikipedia.org/wiki/Svojstvene_vrijednosti_i_svojstveni_vektori" title="Svojstvene vrijednosti i svojstveni vektori – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Svojstvene vrijednosti i svojstveni vektori" data-language-autonym="Srpskohrvatski / српскохрватски" data-language-local-name="Serbo-Croatian" class="interlanguage-link-target">Srpskohrvatski / српскохрватски</a></li><li class="interlanguage-link interwiki-fi mw-list-item"><a href="https://fi.wikipedia.org/wiki/Ominaisarvo,_ominaisvektori_ja_ominaisavaruus" title="Ominaisarvo, ominaisvektori ja ominaisavaruus – Finnish" lang="fi" hreflang="fi" data-title="Ominaisarvo, ominaisvektori ja ominaisavaruus" data-language-autonym="Suomi" data-language-local-name="Finnish" class="interlanguage-link-target">Suomi</a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Egenv%C3%A4rde,_egenvektor_och_egenrum" title="Egenvärde, egenvektor och egenrum – Swedish" lang="sv" hreflang="sv" data-title="Egenvärde, egenvektor och egenrum" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target">Svenska</a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%90%E0%AE%95%E0%AF%86%E0%AE%A9%E0%AF%8D_%E0%AE%AE%E0%AE%A4%E0%AE%BF%E0%AE%AA%E0%AF%8D%E0%AE%AA%E0%AF%81" title="ஐகென் மதிப்பு – Tamil" lang="ta" hreflang="ta" data-title="ஐகென் மதிப்பு" data-language-autonym="தமிழ்" data-language-local-name="Tamil" 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data-language-local-name="Turkish" class="interlanguage-link-target">Türkçe</a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D0%BB%D0%B0%D1%81%D0%BD%D1%96_%D0%B2%D0%B5%D0%BA%D1%82%D0%BE%D1%80%D0%B8_%D1%82%D0%B0_%D0%B2%D0%BB%D0%B0%D1%81%D0%BD%D1%96_%D0%B7%D0%BD%D0%B0%D1%87%D0%B5%D0%BD%D0%BD%D1%8F" title="Власні вектори та власні значення – Ukrainian" lang="uk" hreflang="uk" data-title="Власні вектори та власні значення" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target">Українська</a></li><li class="interlanguage-link interwiki-ur mw-list-item"><a href="https://ur.wikipedia.org/wiki/%D9%88%DB%8C%DA%98%DB%81_%D9%82%D8%AF%D8%B1" title="ویژہ قدر – Urdu" lang="ur" hreflang="ur" data-title="ویژہ قدر" data-language-autonym="اردو" data-language-local-name="Urdu" class="interlanguage-link-target">اردو</a></li><li class="interlanguage-link interwiki-vi mw-list-item"><a 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searchaux" style="display:none">Concepts from linear algebra</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">"Characteristic root" redirects here. For the root of a characteristic equation, see <a href="/wiki/Characteristic_equation_(calculus)" title="Characteristic equation (calculus)">Characteristic equation (calculus)</a>.</div> In <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a>, an eigenvector (<a href="/wiki/Help:IPA/English" title="Help:IPA/English">/ˈaɪɡən-/</a> <a href="/wiki/Help:Pronunciation_respelling_key" title="Help:Pronunciation respelling key">EYE-gən-</a>) or characteristic vector is a <a href="/wiki/Vector_(mathematics_and_physics)" title="Vector (mathematics and physics)">vector</a> that has its <a href="/wiki/Direction_(geometry)" title="Direction (geometry)">direction</a> unchanged (or reversed) by a given <a href="/wiki/Linear_map" title="Linear map">linear transformation</a>. More precisely, an eigenvector, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }">, of a linear transformation, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}">, is <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scaled by a constant factor</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">, when the linear transformation is applied to it: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T\mathbf {v} =\lambda \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T\mathbf {v} =\lambda \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0b55cf369ef5289b52263e955128c2441128ee6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.912ex; height:2.176ex;" alt="{\displaystyle T\mathbf {v} =\lambda \mathbf {v} }">. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> (possibly negative). <a href="/wiki/Euclidean_vector" title="Euclidean vector">Geometrically, vectors</a> are multi-<a href="/wiki/Dimension" title="Dimension">dimensional</a> quantities with magnitude and direction, often pictured as arrows. A linear transformation <a href="/wiki/Rotation_(mathematics)" title="Rotation (mathematics)">rotates</a>, <a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">stretches</a>, or <a href="/wiki/Shear_mapping" title="Shear mapping">shears</a> the vectors upon which it acts. Its eigenvectors are those vectors that are only stretched, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or squished. If the eigenvalue is negative, the eigenvector's direction is reversed.<a href="#cite_note-FOOTNOTEBurdenFaires1993401-1">[1]</a> The eigenvectors and eigenvalues of a linear transformation serve to characterize it, and so they play important roles in all the areas where linear algebra is applied, from <a href="/wiki/Geology" title="Geology">geology</a> to <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>. In particular, it is often the case that a system is represented by a linear transformation whose outputs are fed as inputs to the same transformation (<a href="/wiki/Feedback" title="Feedback">feedback</a>). In such an application, the largest eigenvalue is of particular importance, because it governs the long-term behavior of the system after many applications of the linear transformation, and the associated eigenvector is the <a href="/wiki/Steady_state" title="Steady state">steady state</a> of the system. <style data-mw-deduplicate="TemplateStyles:r886046785">.mw-parser-output .toclimit-2 .toclevel-1 ul,.mw-parser-output .toclimit-3 .toclevel-2 ul,.mw-parser-output .toclimit-4 .toclevel-3 ul,.mw-parser-output .toclimit-5 .toclevel-4 ul,.mw-parser-output .toclimit-6 .toclevel-5 ul,.mw-parser-output .toclimit-7 .toclevel-6 ul{display:none}</style><div class="toclimit-3"><meta property="mw:PageProp/toc" /></div> <div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=1" title="Edit section: Definition">edit</a>]</div> Consider an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n{\times }n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>×</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n{\times }n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/932f7e6e59b498c6d9d918aa98aa9e0ae4697ef9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.598ex; height:1.676ex;" alt="{\displaystyle n{\times }n}"> matrix A and a nonzero vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> of length <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e59df02a9f67a5da3c220f1244c99a46cc4eb1c6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:1.676ex;" alt="{\displaystyle n.}"> If multiplying A with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> (denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3ad03989635ba034915ace643176eb7836efef10" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.154ex; height:2.176ex;" alt="{\displaystyle A\mathbf {v} }">) simply scales <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> by a factor of λ, where λ is a <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> is called an eigenvector of A, and λ is the corresponding eigenvalue. This relationship can be expressed as: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {v} =\lambda \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {v} =\lambda \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1a14308bca76c95b0c802c13524713ae532adb4a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.019ex; height:2.176ex;" alt="{\displaystyle A\mathbf {v} =\lambda \mathbf {v} }">.<a href="#cite_note-2">[2]</a> There is a direct correspondence between n-by-n <a href="/wiki/Square_matrix" title="Square matrix">square matrices</a> and linear transformations from an <a href="/wiki/Dimension_(vector_space)" title="Dimension (vector space)">n-dimensional vector space</a> into itself, given any <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of the vector space. Hence, in a finite-dimensional vector space, it is equivalent to define eigenvalues and eigenvectors using either the language of <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrices</a>, or the language of linear transformations.<a href="#cite_note-FOOTNOTEHerstein1964228,_229-3">[3]</a><a href="#cite_note-FOOTNOTENering197038-4">[4]</a> The following section gives a more general viewpoint that also covers <a href="/wiki/Infinite-dimensional_vector_space" class="mw-redirect" title="Infinite-dimensional vector space">infinite-dimensional vector spaces</a>. <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=2" title="Edit section: Overview">edit</a>]</div> Eigenvalues and eigenvectors feature prominently in the analysis of linear transformations. The prefix <a href="https://en.wiktionary.org/wiki/eigen-" class="extiw" title="wikt:eigen-">eigen-</a> is adopted from the <a href="/wiki/German_language" title="German language">German</a> word <a href="https://en.wiktionary.org/wiki/eigen#German" class="extiw" title="wikt:eigen">eigen</a> (<a href="/wiki/Cognate" title="Cognate">cognate</a> with the <a href="/wiki/English_language" title="English language">English</a> word <a href="https://en.wiktionary.org/wiki/own#English" class="extiw" title="wikt:own">own</a>) for 'proper', 'characteristic', 'own'.<a href="#cite_note-FOOTNOTEBetteridge1965-5">[5]</a><a href="#cite_note-:0-6">[6]</a> Originally used to study <a href="/wiki/Principal_axis_(mechanics)" class="mw-redirect" title="Principal axis (mechanics)">principal axes</a> of the rotational motion of <a href="/wiki/Rigid_body" title="Rigid body">rigid bodies</a>, eigenvalues and eigenvectors have a wide range of applications, for example in <a href="/wiki/Stability_theory" title="Stability theory">stability analysis</a>, <a href="/wiki/Vibration_analysis#eigenvalue_problem" class="mw-redirect" title="Vibration analysis">vibration analysis</a>, <a href="/wiki/Atomic_orbital" title="Atomic orbital">atomic orbitals</a>, <a href="/wiki/Eigenface" title="Eigenface">facial recognition</a>, and <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">matrix diagonalization</a>. In essence, an eigenvector v of a linear transformation T is a nonzero vector that, when T is applied to it, does not change direction. Applying T to the eigenvector only scales the eigenvector by the scalar value λ, called an eigenvalue. This condition can be written as the equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c03cf705786a3d5f73ca46df287ae6739082160" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} ,}"> referred to as the eigenvalue equation or eigenequation. In general, λ may be any <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalar</a>. For example, λ may be negative, in which case the eigenvector reverses direction as part of the scaling, or it may be zero or <a href="/wiki/Complex_number" title="Complex number">complex</a>. <figure typeof="mw:File/Thumb"><a href="/wiki/File:Mona_Lisa_eigenvector_grid.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Mona_Lisa_eigenvector_grid.png/320px-Mona_Lisa_eigenvector_grid.png" decoding="async" width="320" height="223" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Mona_Lisa_eigenvector_grid.png/480px-Mona_Lisa_eigenvector_grid.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3c/Mona_Lisa_eigenvector_grid.png/640px-Mona_Lisa_eigenvector_grid.png 2x" data-file-width="1000" data-file-height="697" /></a><figcaption>In this <a href="/wiki/Shear_mapping" title="Shear mapping">shear mapping</a> the red arrow changes direction, but the blue arrow does not. The blue arrow is an eigenvector of this shear mapping because it does not change direction, and since its length is unchanged, its eigenvalue is 1.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Eigenvectors_of_a_linear_operator.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Eigenvectors_of_a_linear_operator.gif/200px-Eigenvectors_of_a_linear_operator.gif" decoding="async" width="200" height="203" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Eigenvectors_of_a_linear_operator.gif/300px-Eigenvectors_of_a_linear_operator.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f7/Eigenvectors_of_a_linear_operator.gif/400px-Eigenvectors_of_a_linear_operator.gif 2x" data-file-width="480" data-file-height="486" /></a><figcaption>A 2×2 real and symmetric matrix representing a stretching and shearing of the plane. The eigenvectors of the matrix (red lines) are the two special directions such that every point on them will just slide on them.</figcaption></figure> The example here, based on the <a href="/wiki/Mona_Lisa" title="Mona Lisa">Mona Lisa</a>, provides a simple illustration. Each point on the painting can be represented as a vector pointing from the center of the painting to that point. The linear transformation in this example is called a <a href="/wiki/Shear_mapping" title="Shear mapping">shear mapping</a>. Points in the top half are moved to the right, and points in the bottom half are moved to the left, proportional to how far they are from the horizontal axis that goes through the middle of the painting. The vectors pointing to each point in the original image are therefore tilted right or left, and made longer or shorter by the transformation. Points along the horizontal axis do not move at all when this transformation is applied. Therefore, any vector that points directly to the right or left with no vertical component is an eigenvector of this transformation, because the mapping does not change its direction. Moreover, these eigenvectors all have an eigenvalue equal to one, because the mapping does not change their length either. Linear transformations can take many different forms, mapping vectors in a variety of vector spaces, so the eigenvectors can also take many forms. For example, the linear transformation could be a <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> like <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {d}{dx}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {d}{dx}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b9695f938c603b3b40404808946e3c25c6b35b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.636ex; height:3.843ex;" alt="{\displaystyle {\tfrac {d}{dx}}}">, in which case the eigenvectors are functions called <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunctions</a> that are scaled by that differential operator, such as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dx}}e^{\lambda x}=\lambda e^{\lambda x}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>x</mi> </mrow> </msup> <mo>=</mo> <mi>λ</mi> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>x</mi> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dx}}e^{\lambda x}=\lambda e^{\lambda x}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85fcc5dda0cfb276eab35928f9c3ae76cf8578b0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:14.911ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dx}}e^{\lambda x}=\lambda e^{\lambda x}.}"> Alternatively, the linear transformation could take the form of an n by n matrix, in which case the eigenvectors are n by 1 matrices. If the linear transformation is expressed in the form of an n by n matrix A, then the eigenvalue equation for a linear transformation above can be rewritten as the matrix multiplication <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {v} =\lambda \mathbf {v} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {v} =\lambda \mathbf {v} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17f01da2e017cf67290e4b2052126122c1a27934" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.666ex; height:2.509ex;" alt="{\displaystyle A\mathbf {v} =\lambda \mathbf {v} ,}"> where the eigenvector v is an n by 1 matrix. For a matrix, eigenvalues and eigenvectors can be used to <a href="/wiki/Matrix_decomposition" title="Matrix decomposition">decompose the matrix</a>—for example by <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizing</a> it. Eigenvalues and eigenvectors give rise to many closely related mathematical concepts, and the prefix eigen- is applied liberally when naming them: <ul><li>The set of all eigenvectors of a linear transformation, each paired with its corresponding eigenvalue, is called the eigensystem of that transformation.<a href="#cite_note-FOOTNOTEPressTeukolskyVetterlingFlannery2007536-7">[7]</a><a href="#cite_note-FOOTNOTEWolfram.com:_Eigenvector-8">[8]</a></li> <li>The set of all eigenvectors of T corresponding to the same eigenvalue, together with the zero vector, is called an eigenspace, or the characteristic space of T associated with that eigenvalue.<a href="#cite_note-FOOTNOTENering1970107-9">[9]</a></li> <li>If a set of eigenvectors of T forms a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis</a> of the domain of T, then this basis is called an eigenbasis.</li></ul> <div class="mw-heading mw-heading2"><h2 id="History">History</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=3" title="Edit section: History">edit</a>]</div> Eigenvalues are often introduced in the context of <a href="/wiki/Linear_algebra" title="Linear algebra">linear algebra</a> or <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix theory</a>. Historically, however, they arose in the study of <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic forms</a> and <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a>. In the 18th century, <a href="/wiki/Leonhard_Euler" title="Leonhard Euler">Leonhard Euler</a> studied the rotational motion of a <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a>, and discovered the importance of the <a href="/wiki/Moment_of_inertia#Principal_axes" title="Moment of inertia">principal axes</a>.<a href="#cite_note-10">[a]</a> <a href="/wiki/Joseph-Louis_Lagrange" title="Joseph-Louis Lagrange">Joseph-Louis Lagrange</a> realized that the principal axes are the eigenvectors of the inertia matrix.<a href="#cite_note-FOOTNOTEHawkins1975§2-11">[10]</a> In the early 19th century, <a href="/wiki/Augustin-Louis_Cauchy" title="Augustin-Louis Cauchy">Augustin-Louis Cauchy</a> saw how their work could be used to classify the <a href="/wiki/Quadric" title="Quadric">quadric surfaces</a>, and generalized it to arbitrary dimensions.<a href="#cite_note-FOOTNOTEHawkins1975§3-12">[11]</a> Cauchy also coined the term racine caractéristique (characteristic root), for what is now called eigenvalue; his term survives in <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic equation</a>.<a href="#cite_note-13">[b]</a> Later, <a href="/wiki/Joseph_Fourier" title="Joseph Fourier">Joseph Fourier</a> used the work of Lagrange and <a href="/wiki/Pierre-Simon_Laplace" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> to solve the <a href="/wiki/Heat_equation" title="Heat equation">heat equation</a> by <a href="/wiki/Separation_of_variables" title="Separation of variables">separation of variables</a> in his 1822 treatise <a href="/wiki/Joseph_Fourier#The_Analytic_Theory_of_Heat" title="Joseph Fourier">The Analytic Theory of Heat (Théorie analytique de la chaleur)</a>.<a href="#cite_note-FOOTNOTEKline1972p._673-14">[12]</a> <a href="/wiki/Charles-Fran%C3%A7ois_Sturm" class="mw-redirect" title="Charles-François Sturm">Charles-François Sturm</a> elaborated on Fourier's ideas further, and brought them to the attention of Cauchy, who combined them with his own ideas and arrived at the fact that real <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric matrices</a> have real eigenvalues.<a href="#cite_note-FOOTNOTEHawkins1975§3-12">[11]</a> This was extended by <a href="/wiki/Charles_Hermite" title="Charles Hermite">Charles Hermite</a> in 1855 to what are now called <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrices</a>.<a href="#cite_note-FOOTNOTEKline1972pp._807–808-15">[13]</a> Around the same time, <a href="/wiki/Francesco_Brioschi" title="Francesco Brioschi">Francesco Brioschi</a> proved that the eigenvalues of <a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">orthogonal matrices</a> lie on the <a href="/wiki/Unit_circle" title="Unit circle">unit circle</a>,<a href="#cite_note-FOOTNOTEHawkins1975§3-12">[11]</a> and <a href="/wiki/Alfred_Clebsch" title="Alfred Clebsch">Alfred Clebsch</a> found the corresponding result for <a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">skew-symmetric matrices</a>.<a href="#cite_note-FOOTNOTEKline1972pp._807–808-15">[13]</a> Finally, <a href="/wiki/Karl_Weierstrass" title="Karl Weierstrass">Karl Weierstrass</a> clarified an important aspect in the <a href="/wiki/Stability_theory" title="Stability theory">stability theory</a> started by Laplace, by realizing that <a href="/wiki/Defective_matrix" title="Defective matrix">defective matrices</a> can cause instability.<a href="#cite_note-FOOTNOTEHawkins1975§3-12">[11]</a> In the meantime, <a href="/wiki/Joseph_Liouville" title="Joseph Liouville">Joseph Liouville</a> studied eigenvalue problems similar to those of Sturm; the discipline that grew out of their work is now called <a href="/wiki/Sturm%E2%80%93Liouville_theory" title="Sturm–Liouville theory">Sturm–Liouville theory</a>.<a href="#cite_note-FOOTNOTEKline1972pp._715–716-16">[14]</a> <a href="/wiki/Hermann_Schwarz" title="Hermann Schwarz">Schwarz</a> studied the first eigenvalue of <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a> on general domains towards the end of the 19th century, while <a href="/wiki/Henri_Poincar%C3%A9" title="Henri Poincaré">Poincaré</a> studied <a href="/wiki/Poisson%27s_equation" title="Poisson's equation">Poisson's equation</a> a few years later.<a href="#cite_note-FOOTNOTEKline1972pp._706–707-17">[15]</a> At the start of the 20th century, <a href="/wiki/David_Hilbert" title="David Hilbert">David Hilbert</a> studied the eigenvalues of <a href="/wiki/Integral_operator" title="Integral operator">integral operators</a> by viewing the operators as infinite matrices.<a href="#cite_note-FOOTNOTEKline19721063p.-18">[16]</a> He was the first to use the <a href="/wiki/German_language" title="German language">German</a> word eigen, which means "own",<a href="#cite_note-:0-6">[6]</a> to denote eigenvalues and eigenvectors in 1904,<a href="#cite_note-19">[c]</a> though he may have been following a related usage by <a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a>. For some time, the standard term in English was "proper value", but the more distinctive term "eigenvalue" is the standard today.<a href="#cite_note-FOOTNOTEAldrich2006-20">[17]</a> The first numerical algorithm for computing eigenvalues and eigenvectors appeared in 1929, when <a href="/wiki/Richard_von_Mises" title="Richard von Mises">Richard von Mises</a> published the <a href="/wiki/Power_method" class="mw-redirect" title="Power method">power method</a>. One of the most popular methods today, the <a href="/wiki/QR_algorithm" title="QR algorithm">QR algorithm</a>, was proposed independently by <a href="/wiki/John_G._F._Francis" title="John G. F. Francis">John G. F. Francis</a><a href="#cite_note-FOOTNOTEFrancis1961265–271-21">[18]</a> and <a href="/wiki/Vera_Kublanovskaya" title="Vera Kublanovskaya">Vera Kublanovskaya</a><a href="#cite_note-FOOTNOTEKublanovskaya1962-22">[19]</a> in 1961.<a href="#cite_note-FOOTNOTEGolubVan_Loan1996§7.3-23">[20]</a><a href="#cite_note-FOOTNOTEMeyer2000§7.3-24">[21]</a> <div class="mw-heading mw-heading2"><h2 id="Eigenvalues_and_eigenvectors_of_matrices">Eigenvalues and eigenvectors of matrices</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=4" title="Edit section: Eigenvalues and eigenvectors of matrices">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Euclidean_vector" title="Euclidean vector">Euclidean vector</a> and <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix (mathematics)</a></div> Eigenvalues and eigenvectors are often introduced to students in the context of linear algebra courses focused on matrices.<a href="#cite_note-CornellMathCourses-25">[22]</a><a href="#cite_note-UMichMathCourses-26">[23]</a> Furthermore, linear transformations over a finite-dimensional vector space can be represented using matrices,<a href="#cite_note-FOOTNOTEHerstein1964228,_229-3">[3]</a><a href="#cite_note-FOOTNOTENering197038-4">[4]</a> which is especially common in numerical and computational applications.<a href="#cite_note-FOOTNOTEPressTeukolskyVetterlingFlannery200738-27">[24]</a> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Eigenvalue_equation.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Eigenvalue_equation.svg/250px-Eigenvalue_equation.svg.png" decoding="async" width="250" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/58/Eigenvalue_equation.svg/375px-Eigenvalue_equation.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/58/Eigenvalue_equation.svg/500px-Eigenvalue_equation.svg.png 2x" data-file-width="500" data-file-height="400" /></a><figcaption>Matrix A acts by stretching the vector x, not changing its direction, so x is an eigenvector of A.</figcaption></figure> Consider n-dimensional vectors that are formed as a list of n scalars, such as the three-dimensional vectors <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ={\begin{bmatrix}1\\-3\\4\end{bmatrix}}\quad {\mbox{and}}\quad \mathbf {y} ={\begin{bmatrix}-20\\60\\-80\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> <mtr> <mtd> <mn>4</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mtext>and</mtext> </mstyle> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>20</mn> </mtd> </mtr> <mtr> <mtd> <mn>60</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>80</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ={\begin{bmatrix}1\\-3\\4\end{bmatrix}}\quad {\mbox{and}}\quad \mathbf {y} ={\begin{bmatrix}-20\\60\\-80\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de293d0d44d8e19f32243cfaf56a2fa6af43fb30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:32.866ex; height:9.176ex;" alt="{\displaystyle \mathbf {x} ={\begin{bmatrix}1\\-3\\4\end{bmatrix}}\quad {\mbox{and}}\quad \mathbf {y} ={\begin{bmatrix}-20\\60\\-80\end{bmatrix}}.}"> These vectors are said to be <a href="/wiki/Scalar_multiplication" title="Scalar multiplication">scalar multiples</a> of each other, or <a href="/wiki/Parallel_(geometry)" title="Parallel (geometry)">parallel</a> or <a href="/wiki/Collinearity" title="Collinearity">collinear</a>, if there is a scalar λ such that <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} =\lambda \mathbf {y} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} =\lambda \mathbf {y} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9eb82e2821d49cdd3cbc3674e28f06aae77ba270" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.922ex; height:2.509ex;" alt="{\displaystyle \mathbf {x} =\lambda \mathbf {y} .}"> In this case, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =-{\frac {1}{20}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>20</mn> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =-{\frac {1}{20}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d71201b70947b2a10ea46e49bbe6ea13561a818f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.423ex; height:5.176ex;" alt="{\displaystyle \lambda =-{\frac {1}{20}}}">. Now consider the linear transformation of n-dimensional vectors defined by an n by n matrix A, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {v} =\mathbf {w} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {v} =\mathbf {w} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fe15f9b51d050495a3a17711cfd7d797181b271" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.831ex; height:2.509ex;" alt="{\displaystyle A\mathbf {v} =\mathbf {w} ,}"> or <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1n}\\A_{21}&A_{22}&\cdots &A_{2n}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nn}\\\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}={\begin{bmatrix}w_{1}\\w_{2}\\\vdots \\w_{n}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>11</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>12</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>21</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>22</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>n</mi> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> <mtd> <mo>⋱</mo> </mtd> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1n}\\A_{21}&A_{22}&\cdots &A_{2n}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nn}\\\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}={\begin{bmatrix}w_{1}\\w_{2}\\\vdots \\w_{n}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cf37eec75678c8e120c1d0f75448224c2d61bb04" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:41.345ex; height:14.176ex;" alt="{\displaystyle {\begin{bmatrix}A_{11}&A_{12}&\cdots &A_{1n}\\A_{21}&A_{22}&\cdots &A_{2n}\\\vdots &\vdots &\ddots &\vdots \\A_{n1}&A_{n2}&\cdots &A_{nn}\\\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\\\vdots \\v_{n}\end{bmatrix}}={\begin{bmatrix}w_{1}\\w_{2}\\\vdots \\w_{n}\end{bmatrix}}}"> where, for each row, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w_{i}=A_{i1}v_{1}+A_{i2}v_{2}+\cdots +A_{in}v_{n}=\sum _{j=1}^{n}A_{ij}v_{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>w</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>1</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mn>2</mn> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>n</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle w_{i}=A_{i1}v_{1}+A_{i2}v_{2}+\cdots +A_{in}v_{n}=\sum _{j=1}^{n}A_{ij}v_{j}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08d123f33013c06fe878cd5bcf60a3e967a7186a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:46.52ex; height:7.176ex;" alt="{\displaystyle w_{i}=A_{i1}v_{1}+A_{i2}v_{2}+\cdots +A_{in}v_{n}=\sum _{j=1}^{n}A_{ij}v_{j}.}"> If it occurs that v and w are scalar multiples, that is if <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {v} =\mathbf {w} =\lambda \mathbf {v} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">w</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {v} =\mathbf {w} =\lambda \mathbf {v} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dd1efab8aaff11ec360ed460c28b231bdf53dc6c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.695ex; height:2.509ex;" alt="{\displaystyle A\mathbf {v} =\mathbf {w} =\lambda \mathbf {v} ,}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(1)</td></tr></tbody></table> then v is an eigenvector of the linear transformation A and the scale factor λ is the eigenvalue corresponding to that eigenvector. Equation (<a href="#math_1">1</a>) is the eigenvalue equation for the matrix A. Equation (<a href="#math_1">1</a>) can be stated equivalently as <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25d526099011bbd1f25ca18d062053b519c4ca1c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.8ex; height:2.843ex;" alt="{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} ,}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(2)</td></tr></tbody></table> where I is the n by n <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> and 0 is the zero vector. <div class="mw-heading mw-heading3"><h3 id="Eigenvalues_and_the_characteristic_polynomial">Eigenvalues and the characteristic polynomial</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=5" title="Edit section: Eigenvalues and the characteristic polynomial">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">Characteristic polynomial</a></div> Equation (<a href="#math_2">2</a>) has a nonzero solution v <a href="/wiki/If_and_only_if" title="If and only if">if and only if</a> the <a href="/wiki/Determinant" title="Determinant">determinant</a> of the matrix (A − λI) is zero. Therefore, the eigenvalues of A are values of λ that satisfy the equation <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A-\lambda I)=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A-\lambda I)=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/592cb586250daeb4721c83da10cc2b812fe98dcd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.41ex; height:2.843ex;" alt="{\displaystyle \det(A-\lambda I)=0}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(3)</td></tr></tbody></table> Using the <a href="/wiki/Leibniz_formula_for_determinants" title="Leibniz formula for determinants">Leibniz formula for determinants</a>, the left-hand side of equation (<a href="#math_3">3</a>) is a <a href="/wiki/Polynomial" title="Polynomial">polynomial</a> function of the variable λ and the <a href="/wiki/Degree_of_a_polynomial" title="Degree of a polynomial">degree</a> of this polynomial is n, the order of the matrix A. Its <a href="/wiki/Coefficient" title="Coefficient">coefficients</a> depend on the entries of A, except that its term of degree n is always (−1)nλn. This polynomial is called the <a href="/wiki/Characteristic_polynomial" title="Characteristic polynomial">characteristic polynomial</a> of A. Equation (<a href="#math_3">3</a>) is called the characteristic equation or the secular equation of A. The <a href="/wiki/Fundamental_theorem_of_algebra" title="Fundamental theorem of algebra">fundamental theorem of algebra</a> implies that the characteristic polynomial of an n-by-n matrix A, being a polynomial of degree n, can be <a href="/wiki/Factorization" title="Factorization">factored</a> into the product of n linear terms, <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )(\lambda _{2}-\lambda )\cdots (\lambda _{n}-\lambda ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>⋯</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )(\lambda _{2}-\lambda )\cdots (\lambda _{n}-\lambda ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/978b7b557de455284742a2172d1fb845f5ca3e8d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:44.799ex; height:2.843ex;" alt="{\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )(\lambda _{2}-\lambda )\cdots (\lambda _{n}-\lambda ),}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(4)</td></tr></tbody></table> where each λi may be real but in general is a complex number. The numbers λ1, λ2, ..., λn, which may not all have distinct values, are roots of the polynomial and are the eigenvalues of A. As a brief example, which is described in more detail in the examples section later, consider the matrix <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b50d689a4b5e815e14a9afb3b8da06fd6cc8e723" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.342ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}"> Taking the determinant of (A − λI), the characteristic polynomial of A is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}=3-4\lambda +\lambda ^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <mo>−</mo> <mi>λ</mi> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> <mo>−</mo> <mi>λ</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mn>4</mn> <mi>λ</mi> <mo>+</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}=3-4\lambda +\lambda ^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ba23d3efcb8d6100a9128f91c92cd167b1ca24f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.847ex; height:6.176ex;" alt="{\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}=3-4\lambda +\lambda ^{2}.}"> Setting the characteristic polynomial equal to zero, it has roots at λ=1 and λ=3, which are the two eigenvalues of A. The eigenvectors corresponding to each eigenvalue can be found by solving for the components of v in the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/008c8f5f80ae7e23df86c574ad9a8ebbbfc9a88f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.153ex; height:2.843ex;" alt="{\displaystyle \left(A-\lambda I\right)\mathbf {v} =\mathbf {0} }">. In this example, the eigenvectors are any nonzero scalar multiples of <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {v} _{\lambda =3}={\begin{bmatrix}1\\1\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>=</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {v} _{\lambda =3}={\begin{bmatrix}1\\1\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/060d71942a0e2d5bf9a9b585436b610e42576622" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.15ex; height:6.176ex;" alt="{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}1\\-1\end{bmatrix}},\quad \mathbf {v} _{\lambda =3}={\begin{bmatrix}1\\1\end{bmatrix}}.}"> If the entries of the matrix A are all real numbers, then the coefficients of the characteristic polynomial will also be real numbers, but the eigenvalues may still have nonzero imaginary parts. The entries of the corresponding eigenvectors therefore may also have nonzero imaginary parts. Similarly, the eigenvalues may be <a href="/wiki/Irrational_number" title="Irrational number">irrational numbers</a> even if all the entries of A are <a href="/wiki/Rational_number" title="Rational number">rational numbers</a> or even if they are all integers. However, if the entries of A are all <a href="/wiki/Algebraic_number" title="Algebraic number">algebraic numbers</a>, which include the rationals, the eigenvalues must also be algebraic numbers. The non-real roots of a real polynomial with real coefficients can be grouped into pairs of <a href="/wiki/Complex_conjugate" title="Complex conjugate">complex conjugates</a>, namely with the two members of each pair having imaginary parts that differ only in sign and the same real part. If the degree is odd, then by the <a href="/wiki/Intermediate_value_theorem" title="Intermediate value theorem">intermediate value theorem</a> at least one of the roots is real. Therefore, any <a href="/wiki/Real_matrix" class="mw-redirect" title="Real matrix">real matrix</a> with odd order has at least one real eigenvalue, whereas a real matrix with even order may not have any real eigenvalues. The eigenvectors associated with these complex eigenvalues are also complex and also appear in complex conjugate pairs. <div class="mw-heading mw-heading3"><h3 id="Spectrum_of_a_matrix">Spectrum of a matrix</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=6" title="Edit section: Spectrum of a matrix">edit</a>]</div> The <a href="/wiki/Spectrum_of_a_matrix" title="Spectrum of a matrix">spectrum</a> of a matrix is the list of eigenvalues, repeated according to multiplicity; in an alternative notation the set of eigenvalues with their multiplicities. An important quantity associated with the spectrum is the maximum absolute value of any eigenvalue. This is known as the <a href="/wiki/Spectral_radius#Matrices" title="Spectral radius">spectral radius</a> of the matrix. <div class="mw-heading mw-heading3"><h3 id="Algebraic_multiplicity">Algebraic multiplicity</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=7" title="Edit section: Algebraic multiplicity">edit</a>]</div> Let λi be an eigenvalue of an n by n matrix A. The algebraic multiplicity μA(λi) of the eigenvalue is its <a href="/wiki/Multiple_roots_of_a_polynomial" class="mw-redirect" title="Multiple roots of a polynomial">multiplicity as a root</a> of the characteristic polynomial, that is, the largest integer k such that (λ − λi)k <a href="/wiki/Polynomial_division" class="mw-redirect" title="Polynomial division">divides evenly</a> that polynomial.<a href="#cite_note-FOOTNOTENering1970107-9">[9]</a><a href="#cite_note-FOOTNOTEFraleigh1976358-28">[25]</a><a href="#cite_note-FOOTNOTEGolubVan_Loan1996316-29">[26]</a> Suppose a matrix A has dimension n and d ≤ n distinct eigenvalues. Whereas equation (<a href="#math_4">4</a>) factors the characteristic polynomial of A into the product of n linear terms with some terms potentially repeating, the characteristic polynomial can also be written as the product of d terms each corresponding to a distinct eigenvalue and raised to the power of the algebraic multiplicity, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )^{\mu _{A}(\lambda _{1})}(\lambda _{2}-\lambda )^{\mu _{A}(\lambda _{2})}\cdots (\lambda _{d}-\lambda )^{\mu _{A}(\lambda _{d})}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>⋯</mo> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo>−</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )^{\mu _{A}(\lambda _{1})}(\lambda _{2}-\lambda )^{\mu _{A}(\lambda _{2})}\cdots (\lambda _{d}-\lambda )^{\mu _{A}(\lambda _{d})}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7801030a602a34d170993ac2595ac17636337521" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:61.076ex; height:3.343ex;" alt="{\displaystyle \det(A-\lambda I)=(\lambda _{1}-\lambda )^{\mu _{A}(\lambda _{1})}(\lambda _{2}-\lambda )^{\mu _{A}(\lambda _{2})}\cdots (\lambda _{d}-\lambda )^{\mu _{A}(\lambda _{d})}.}"> If d = n then the right-hand side is the product of n linear terms and this is the same as equation (<a href="#math_4">4</a>). The size of each eigenvalue's algebraic multiplicity is related to the dimension n as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}1&\leq \mu _{A}(\lambda _{i})\leq n,\\\mu _{A}&=\sum _{i=1}^{d}\mu _{A}\left(\lambda _{i}\right)=n.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi></mi> <mo>≤</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>n</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </munderover> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mrow> <mo>(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mi>n</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}1&\leq \mu _{A}(\lambda _{i})\leq n,\\\mu _{A}&=\sum _{i=1}^{d}\mu _{A}\left(\lambda _{i}\right)=n.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9175a685ce1989db4c4d160fcedcc69e68d58115" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:22.816ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}1&\leq \mu _{A}(\lambda _{i})\leq n,\\\mu _{A}&=\sum _{i=1}^{d}\mu _{A}\left(\lambda _{i}\right)=n.\end{aligned}}}"> If μA(λi) = 1, then λi is said to be a simple eigenvalue.<a href="#cite_note-FOOTNOTEGolubVan_Loan1996316-29">[26]</a> If μA(λi) equals the geometric multiplicity of λi, γA(λi), defined in the next section, then λi is said to be a semisimple eigenvalue. <div class="mw-heading mw-heading3"><h3 id="Eigenspaces,_geometric_multiplicity,_and_the_eigenbasis_for_matrices">Eigenspaces, geometric multiplicity, and the eigenbasis for matrices</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=8" title="Edit section: Eigenspaces, geometric multiplicity, and the eigenbasis for matrices">edit</a>]</div> Given a particular eigenvalue λ of the n by n matrix A, define the <a href="/wiki/Set_(mathematics)" title="Set (mathematics)">set</a> E to be all vectors v that satisfy equation (<a href="#math_2">2</a>), <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\left\{\mathbf {v} :\left(A-\lambda I\right)\mathbf {v} =\mathbf {0} \right\}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>:</mo> <mrow> <mo>(</mo> <mrow> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mn mathvariant="bold">0</mn> </mrow> </mrow> <mo>}</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\left\{\mathbf {v} :\left(A-\lambda I\right)\mathbf {v} =\mathbf {0} \right\}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f98edd9186f222b41c08d1c86ebe3272af0c977b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.734ex; height:2.843ex;" alt="{\displaystyle E=\left\{\mathbf {v} :\left(A-\lambda I\right)\mathbf {v} =\mathbf {0} \right\}.}"> On one hand, this set is precisely the <a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">kernel</a> or nullspace of the matrix (A − λI). On the other hand, by definition, any nonzero vector that satisfies this condition is an eigenvector of A associated with λ. So, the set E is the <a href="/wiki/Union_(set_theory)" title="Union (set theory)">union</a> of the zero vector with the set of all eigenvectors of A associated with λ, and E equals the nullspace of (A − λI). E is called the eigenspace or characteristic space of A associated with λ.<a href="#cite_note-FOOTNOTEAnton1987305,_307-30">[27]</a><a href="#cite_note-FOOTNOTENering1970107-9">[9]</a> In general λ is a complex number and the eigenvectors are complex n by 1 matrices. A property of the nullspace is that it is a <a href="/wiki/Linear_subspace" title="Linear subspace">linear subspace</a>, so E is a linear subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}">. Because the eigenspace E is a linear subspace, it is <a href="/wiki/Closure_(mathematics)" title="Closure (mathematics)">closed</a> under addition. That is, if two vectors u and v belong to the set E, written u, v ∈ E, then (u + v) ∈ E or equivalently A(u + v) = λ(u + v). This can be checked using the <a href="/wiki/Distributive_property" title="Distributive property">distributive property</a> of matrix multiplication. Similarly, because E is a linear subspace, it is closed under scalar multiplication. That is, if v ∈ E and α is a complex number, (αv) ∈ E or equivalently A(αv) = λ(αv). This can be checked by noting that multiplication of complex matrices by complex numbers is <a href="/wiki/Commutative_property" title="Commutative property">commutative</a>. As long as u + v and αv are not zero, they are also eigenvectors of A associated with λ. The dimension of the eigenspace E associated with λ, or equivalently the maximum number of linearly independent eigenvectors associated with λ, is referred to as the eigenvalue's geometric multiplicity <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{A}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{A}(\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca33c07b96d989a31433d2d0536b16ec930ee0c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.834ex; height:2.843ex;" alt="{\displaystyle \gamma _{A}(\lambda )}">. Because E is also the nullspace of (A − λI), the geometric multiplicity of λ is the dimension of the nullspace of (A − λI), also called the nullity of (A − λI), which relates to the dimension and rank of (A − λI) as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{A}(\lambda )=n-\operatorname {rank} (A-\lambda I).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>n</mi> <mo>−</mo> <mi>rank</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{A}(\lambda )=n-\operatorname {rank} (A-\lambda I).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d05ef33a094de7bd684bbe6833c58fd9c287b749" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:27.328ex; height:2.843ex;" alt="{\displaystyle \gamma _{A}(\lambda )=n-\operatorname {rank} (A-\lambda I).}"> Because of the definition of eigenvalues and eigenvectors, an eigenvalue's geometric multiplicity must be at least one, that is, each eigenvalue has at least one associated eigenvector. Furthermore, an eigenvalue's geometric multiplicity cannot exceed its algebraic multiplicity. Additionally, recall that an eigenvalue's algebraic multiplicity cannot exceed n. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\leq \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>≤</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\leq \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e10188857eb5dcd123f4a50bbe973b2dedc9a923" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.717ex; height:2.843ex;" alt="{\displaystyle 1\leq \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )\leq n}"> To prove the inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f88a7977850f2a658b3f3d3e0e80fe63d4addb7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.963ex; height:2.843ex;" alt="{\displaystyle \gamma _{A}(\lambda )\leq \mu _{A}(\lambda )}">, consider how the definition of geometric multiplicity implies the existence of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{A}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{A}(\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca33c07b96d989a31433d2d0536b16ec930ee0c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.834ex; height:2.843ex;" alt="{\displaystyle \gamma _{A}(\lambda )}"> <a href="/wiki/Orthonormality" title="Orthonormality">orthonormal</a> eigenvectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {v}}_{1},\,\ldots ,\,{\boldsymbol {v}}_{\gamma _{A}(\lambda )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {v}}_{1},\,\ldots ,\,{\boldsymbol {v}}_{\gamma _{A}(\lambda )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a28d71457b1b1fe976c59d7d6757b4331dc1089" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:14.129ex; height:2.509ex;" alt="{\displaystyle {\boldsymbol {v}}_{1},\,\ldots ,\,{\boldsymbol {v}}_{\gamma _{A}(\lambda )}}">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\boldsymbol {v}}_{k}=\lambda {\boldsymbol {v}}_{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mi>λ</mi> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\boldsymbol {v}}_{k}=\lambda {\boldsymbol {v}}_{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99bcab59336bde66c2cbce8f5fabd12436792d44" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.01ex; height:2.509ex;" alt="{\displaystyle A{\boldsymbol {v}}_{k}=\lambda {\boldsymbol {v}}_{k}}">. We can therefore find a (unitary) matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> whose first <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{A}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{A}(\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca33c07b96d989a31433d2d0536b16ec930ee0c7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.834ex; height:2.843ex;" alt="{\displaystyle \gamma _{A}(\lambda )}"> columns are these eigenvectors, and whose remaining columns can be any orthonormal set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n-\gamma _{A}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>−</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n-\gamma _{A}(\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e73eebe39bd0f4a10b8e8ebf007616f5f6f0804c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.069ex; height:2.843ex;" alt="{\displaystyle n-\gamma _{A}(\lambda )}"> vectors orthogonal to these eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. Then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}"> has full rank and is therefore invertible. Evaluating <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D:=V^{T}AV}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>:=</mo> <msup> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>T</mi> </mrow> </msup> <mi>A</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D:=V^{T}AV}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e6dc7e8ad04ecc6d9c0073f68f44122e2442459c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.506ex; height:2.676ex;" alt="{\displaystyle D:=V^{T}AV}">, we get a matrix whose top left block is the diagonal matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda I_{\gamma _{A}(\lambda )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <msub> <mi>I</mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda I_{\gamma _{A}(\lambda )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/200d2f031efc3d7effd009905f264f78e237ee02" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.865ex; height:3.009ex;" alt="{\displaystyle \lambda I_{\gamma _{A}(\lambda )}}">. This can be seen by evaluating what the left-hand side does to the first column basis vectors. By reorganizing and adding <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -\xi V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mi>ξ</mi> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -\xi V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ced6d478004a248184138af3da8fcb21671e4d5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.625ex; height:2.509ex;" alt="{\displaystyle -\xi V}"> on both sides, we get <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A-\xi I)V=V(D-\xi I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>ξ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mi>V</mi> <mo>=</mo> <mi>V</mi> <mo stretchy="false">(</mo> <mi>D</mi> <mo>−</mo> <mi>ξ</mi> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A-\xi I)V=V(D-\xi I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07d1f3af51f0dda6d9080c033b80b7364e84c232" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.043ex; height:2.843ex;" alt="{\displaystyle (A-\xi I)V=V(D-\xi I)}"> since <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> commutes with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}">. In other words, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A-\xi I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>−</mo> <mi>ξ</mi> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A-\xi I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/116784e14f43f7644c306dc7eac13d281e8351ba" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.785ex; height:2.509ex;" alt="{\displaystyle A-\xi I}"> is similar to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D-\xi I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>−</mo> <mi>ξ</mi> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D-\xi I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3280e4c52d6e32e5f10336524348c494f522a930" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.966ex; height:2.509ex;" alt="{\displaystyle D-\xi I}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A-\xi I)=\det(D-\xi I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>ξ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>−</mo> <mi>ξ</mi> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A-\xi I)=\det(D-\xi I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd07a91da429f978ec368accf8961c7fc180a2cd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.928ex; height:2.843ex;" alt="{\displaystyle \det(A-\xi I)=\det(D-\xi I)}">. But from the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}">, we know that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(D-\xi I)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo>−</mo> <mi>ξ</mi> <mi>I</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(D-\xi I)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e54fb904e76ca871c917f6052cc087bb606c600" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.005ex; height:2.843ex;" alt="{\displaystyle \det(D-\xi I)}"> contains a factor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\xi -\lambda )^{\gamma _{A}(\lambda )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>ξ</mi> <mo>−</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\xi -\lambda )^{\gamma _{A}(\lambda )}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3f9ff0335564c3ad1ecf612dda97c8541de2cf6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.521ex; height:3.343ex;" alt="{\displaystyle (\xi -\lambda )^{\gamma _{A}(\lambda )}}">, which means that the algebraic multiplicity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }"> must satisfy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{A}(\lambda )\geq \gamma _{A}(\lambda )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>λ</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{A}(\lambda )\geq \gamma _{A}(\lambda )}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fd457b5f339994ebc2b19c5a1ca4e13ab3ab216b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.963ex; height:2.843ex;" alt="{\displaystyle \mu _{A}(\lambda )\geq \gamma _{A}(\lambda )}">. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> has <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d\leq n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>d</mi> <mo>≤</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle d\leq n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/955b752dac67ec95b171a3a97b5de7a0ed5a7711" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.709ex; height:2.343ex;" alt="{\displaystyle d\leq n}"> distinct eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\ldots ,\lambda _{d}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\ldots ,\lambda _{d}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/09fc600abaee26294eb8f72809cf2fce4cd7ba80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.035ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\ldots ,\lambda _{d}}">, where the geometric multiplicity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}"> is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{A}(\lambda _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{A}(\lambda _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5217c911493cde199c956ba3bff54152e8e148dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.633ex; height:2.843ex;" alt="{\displaystyle \gamma _{A}(\lambda _{i})}">. The total geometric multiplicity of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\gamma _{A}&=\sum _{i=1}^{d}\gamma _{A}(\lambda _{i}),\\d&\leq \gamma _{A}\leq n,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>d</mi> </mrow> </munderover> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>d</mi> </mtd> <mtd> <mi></mi> <mo>≤</mo> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>≤</mo> <mi>n</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\gamma _{A}&=\sum _{i=1}^{d}\gamma _{A}(\lambda _{i}),\\d&\leq \gamma _{A}\leq n,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba4a033988e103092c89f6f5f88634a856a31d0f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:17.541ex; height:10.509ex;" alt="{\displaystyle {\begin{aligned}\gamma _{A}&=\sum _{i=1}^{d}\gamma _{A}(\lambda _{i}),\\d&\leq \gamma _{A}\leq n,\end{aligned}}}"> is the dimension of the <a href="/wiki/Linear_subspace#Sum" title="Linear subspace">sum</a> of all the eigenspaces of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">'s eigenvalues, or equivalently the maximum number of linearly independent eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{A}=n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo>=</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{A}=n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b023958dcfb98e7e0c448ee758bb96d99bec1ddf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.162ex; height:2.176ex;" alt="{\displaystyle \gamma _{A}=n}">, then <ul><li>The direct sum of the eigenspaces of all of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">'s eigenvalues is the entire vector space <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}">.</li> <li>A basis of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"> can be formed from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> linearly independent eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">; such a basis is called an eigenbasis</li> <li>Any vector in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {C} ^{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {C} ^{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.897ex; height:2.343ex;" alt="{\displaystyle \mathbb {C} ^{n}}"> can be written as a linear combination of eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Additional_properties">Additional properties</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=9" title="Edit section: Additional properties">edit</a>]</div> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> be an arbitrary <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrix of complex numbers with eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\ldots ,\lambda _{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\ldots ,\lambda _{n}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a896f82d292f2489a979c7a2c7a52561df77dd4d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.161ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\ldots ,\lambda _{n}}">. Each eigenvalue appears <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{A}(\lambda _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{A}(\lambda _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4caeddaf6798f1d034ec6fb21c8a0e2dd938ec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.831ex; height:2.843ex;" alt="{\displaystyle \mu _{A}(\lambda _{i})}"> times in this list, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{A}(\lambda _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>A</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{A}(\lambda _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4caeddaf6798f1d034ec6fb21c8a0e2dd938ec2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.831ex; height:2.843ex;" alt="{\displaystyle \mu _{A}(\lambda _{i})}"> is the eigenvalue's algebraic multiplicity. The following are properties of this matrix and its eigenvalues: <ul><li>The <a href="/wiki/Trace_(linear_algebra)" title="Trace (linear algebra)">trace</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, defined as the sum of its diagonal elements, is also the sum of all eigenvalues,<a href="#cite_note-FOOTNOTEBeauregardFraleigh1973307-31">[28]</a><a href="#cite_note-FOOTNOTEHerstein1964272-32">[29]</a><a href="#cite_note-FOOTNOTENering1970115–116-33">[30]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {tr} (A)=\sum _{i=1}^{n}a_{ii}=\sum _{i=1}^{n}\lambda _{i}=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>tr</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {tr} (A)=\sum _{i=1}^{n}a_{ii}=\sum _{i=1}^{n}\lambda _{i}=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/14f5065ea811360e42eed59665d54f30db38c1b8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:46.184ex; height:6.843ex;" alt="{\displaystyle \operatorname {tr} (A)=\sum _{i=1}^{n}a_{ii}=\sum _{i=1}^{n}\lambda _{i}=\lambda _{1}+\lambda _{2}+\cdots +\lambda _{n}.}"></dd></dl></li> <li>The <a href="/wiki/Determinant" title="Determinant">determinant</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is the product of all its eigenvalues,<a href="#cite_note-FOOTNOTEBeauregardFraleigh1973307-31">[28]</a><a href="#cite_note-FOOTNOTEHerstein1964290-34">[31]</a><a href="#cite_note-FOOTNOTENering1970116-35">[32]</a> <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> <mo>=</mo> <munderover> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </munderover> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋯</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42699557860330ebb8293b3aef3d8a5b10461ff1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:30.027ex; height:6.843ex;" alt="{\displaystyle \det(A)=\prod _{i=1}^{n}\lambda _{i}=\lambda _{1}\lambda _{2}\cdots \lambda _{n}.}"></dd></dl></li> <li>The eigenvalues of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">th power of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">; i.e., the eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/635d375f6533a2afe158ada75a3ebb56d623965d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.832ex; height:2.676ex;" alt="{\displaystyle A^{k}}">, for any positive integer <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">, are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}^{k},\ldots ,\lambda _{n}^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}^{k},\ldots ,\lambda _{n}^{k}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6926093affc360d167efcb541f640fd5bbebe091" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.196ex; height:3.176ex;" alt="{\displaystyle \lambda _{1}^{k},\ldots ,\lambda _{n}^{k}}">.</li> <li>The matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> if and only if every eigenvalue is nonzero.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is invertible, then the eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83ba3a7118652cffd5de466dc439ee9184371d50" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.076ex; height:2.676ex;" alt="{\displaystyle A^{-1}}"> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle {\frac {1}{\lambda _{1}}},\ldots ,{\frac {1}{\lambda _{n}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mfrac> </mrow> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle {\frac {1}{\lambda _{1}}},\ldots ,{\frac {1}{\lambda _{n}}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b32fe9a025a80d01ab0ea5d958dbb848f6466bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:10.564ex; height:4.009ex;" alt="{\textstyle {\frac {1}{\lambda _{1}}},\ldots ,{\frac {1}{\lambda _{n}}}}"> and each eigenvalue's geometric multiplicity coincides. Moreover, since the characteristic polynomial of the inverse is the <a href="/wiki/Reciprocal_polynomial" title="Reciprocal polynomial">reciprocal polynomial</a> of the original, the eigenvalues share the same algebraic multiplicity.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is equal to its <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44e23745a51c2c2d8d91fd98c1cf721573747ece" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.797ex; height:2.343ex;" alt="{\displaystyle A^{*}}">, or equivalently if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a>, then every eigenvalue is real. The same is true of any <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> real matrix.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is not only Hermitian but also <a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">positive-definite</a>, positive-semidefinite, negative-definite, or negative-semidefinite, then every eigenvalue is positive, non-negative, negative, or non-positive, respectively.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is <a href="/wiki/Unitary_matrix" title="Unitary matrix">unitary</a>, every eigenvalue has absolute value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\lambda _{i}|=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\lambda _{i}|=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f55de2fc7c74d71e685c0d47390e81f11a38e7f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.71ex; height:2.843ex;" alt="{\displaystyle |\lambda _{i}|=1}">.</li> <li>If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrix and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{1},\ldots ,\lambda _{k}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lambda _{1},\ldots ,\lambda _{k}\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2e60b2ee8349151ab891ab98fb0cf2e6d516484" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.357ex; height:2.843ex;" alt="{\displaystyle \{\lambda _{1},\ldots ,\lambda _{k}\}}"> are its eigenvalues, then the eigenvalues of matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I+A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>+</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I+A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2346f1d10f8965bc8c2f4b5444655642fb5f030a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.755ex; height:2.343ex;" alt="{\displaystyle I+A}"> (where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/535ea7fc4134a31cbe2251d9d3511374bc41be9f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}"> is the identity matrix) are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{1}+1,\ldots ,\lambda _{k}+1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lambda _{1}+1,\ldots ,\lambda _{k}+1\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11ed0826f715a95967bd69617e8484805c1c3a88" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.362ex; height:2.843ex;" alt="{\displaystyle \{\lambda _{1}+1,\ldots ,\lambda _{k}+1\}}">. Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha \in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c2956c000d6b14cfab5918e7129edd52af31706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.006ex; height:2.176ex;" alt="{\displaystyle \alpha \in \mathbb {C} }">, the eigenvalues of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha I+A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>α</mi> <mi>I</mi> <mo>+</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha I+A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a14adb743f510153c21e9ad28443be35f54808" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.243ex; height:2.343ex;" alt="{\displaystyle \alpha I+A}"> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\lambda _{1}+\alpha ,\ldots ,\lambda _{k}+\alpha \}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mi>α</mi> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>α</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\lambda _{1}+\alpha ,\ldots ,\lambda _{k}+\alpha \}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ca281a92a0f84646e8f669fb74530c0a11a1d7c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.013ex; height:2.843ex;" alt="{\displaystyle \{\lambda _{1}+\alpha ,\ldots ,\lambda _{k}+\alpha \}}">. More generally, for a polynomial <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"> the eigenvalues of matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P(A)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> <mo stretchy="false">(</mo> <mi>A</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P(A)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4f264d19e21604793c6dc54f8044df454db82744" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.298ex; height:2.843ex;" alt="{\displaystyle P(A)}"> are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{P(\lambda _{1}),\ldots ,P(\lambda _{k})\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>P</mi> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{P(\lambda _{1}),\ldots ,P(\lambda _{k})\}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38df6e71f258cb42b9dd0c32fe69ce0dc7ee0db0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.466ex; height:2.843ex;" alt="{\displaystyle \{P(\lambda _{1}),\ldots ,P(\lambda _{k})\}}">.</li></ul> <div class="mw-heading mw-heading3"><h3 id="Left_and_right_eigenvectors">Left and right eigenvectors</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=10" title="Edit section: Left and right eigenvectors">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Left_and_right_(algebra)" title="Left and right (algebra)">left and right (algebra)</a></div> Many disciplines traditionally represent vectors as matrices with a single column rather than as matrices with a single row. For that reason, the word "eigenvector" in the context of matrices almost always refers to a right eigenvector, namely a column vector that right multiplies the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> in the defining equation, equation (<a href="#math_1">1</a>), <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A\mathbf {v} =\lambda \mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A\mathbf {v} =\lambda \mathbf {v} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bab09470c0a743ad21485fe638ce04b8fdf3e68c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.666ex; height:2.176ex;" alt="{\displaystyle A\mathbf {v} =\lambda \mathbf {v} .}"> The eigenvalue and eigenvector problem can also be defined for row vectors that left multiply matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">. In this formulation, the defining equation is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} A=\kappa \mathbf {u} ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mi>A</mi> <mo>=</mo> <mi>κ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} A=\kappa \mathbf {u} ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bc646b147f57577a121c299aaa300c1dd928bff6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.798ex; height:2.509ex;" alt="{\displaystyle \mathbf {u} A=\kappa \mathbf {u} ,}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"> is a scalar and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> is a <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0bce5f6a6d0d32834484048c16f3b39f9c23d076" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.398ex; height:2.176ex;" alt="{\displaystyle 1\times n}"> matrix. Any row vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>u</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle u}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}"> satisfying this equation is called a left eigenvector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>κ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \kappa }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54ddec2e922c5caea4e47d04feef86e782dc8e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:1.676ex;" alt="{\displaystyle \kappa }"> is its associated eigenvalue. Taking the transpose of this equation, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\textsf {T}}\mathbf {u} ^{\textsf {T}}=\kappa \mathbf {u} ^{\textsf {T}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>=</mo> <mi>κ</mi> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\textsf {T}}\mathbf {u} ^{\textsf {T}}=\kappa \mathbf {u} ^{\textsf {T}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0fc4d00ecb8d5fdac2710d2e4156b66214ba3f5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.852ex; height:2.676ex;" alt="{\displaystyle A^{\textsf {T}}\mathbf {u} ^{\textsf {T}}=\kappa \mathbf {u} ^{\textsf {T}}.}"> Comparing this equation to equation (<a href="#math_1">1</a>), it follows immediately that a left eigenvector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> is the same as the transpose of a right eigenvector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a374264c83956baa6d88ea2b8d55aeaafc86b8a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle A^{\textsf {T}}}">, with the same eigenvalue. Furthermore, since the characteristic polynomial of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a374264c83956baa6d88ea2b8d55aeaafc86b8a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.676ex;" alt="{\displaystyle A^{\textsf {T}}}"> is the same as the characteristic polynomial of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, the left and right eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> are associated with the same eigenvalues. <div class="mw-heading mw-heading3"><h3 id="Diagonalization_and_the_eigendecomposition">Diagonalization and the eigendecomposition</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=11" title="Edit section: Diagonalization and the eigendecomposition">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">Eigendecomposition of a matrix</a></div> Suppose the eigenvectors of A form a basis, or equivalently A has n linearly independent eigenvectors v1, v2, ..., vn with associated eigenvalues λ1, λ2, ..., λn. The eigenvalues need not be distinct. Define a <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> Q whose columns are the n linearly independent eigenvectors of A, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q={\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q={\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1e72813277b39ec5350bd43c1dd4a697e46b6e59" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.88ex; height:2.843ex;" alt="{\displaystyle Q={\begin{bmatrix}\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{n}\end{bmatrix}}.}"></dd></dl> Since each column of Q is an eigenvector of A, right multiplying A by Q scales each column of Q by its associated eigenvalue, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AQ={\begin{bmatrix}\lambda _{1}\mathbf {v} _{1}&\lambda _{2}\mathbf {v} _{2}&\cdots &\lambda _{n}\mathbf {v} _{n}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>Q</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AQ={\begin{bmatrix}\lambda _{1}\mathbf {v} _{1}&\lambda _{2}\mathbf {v} _{2}&\cdots &\lambda _{n}\mathbf {v} _{n}\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/054311dce4186cac53a7af3e4dc9e6167a852228" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.015ex; height:2.843ex;" alt="{\displaystyle AQ={\begin{bmatrix}\lambda _{1}\mathbf {v} _{1}&\lambda _{2}\mathbf {v} _{2}&\cdots &\lambda _{n}\mathbf {v} _{n}\end{bmatrix}}.}"></dd></dl> With this in mind, define a diagonal matrix Λ where each diagonal element Λii is the eigenvalue associated with the ith column of Q. Then <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AQ=Q\Lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>Q</mi> <mo>=</mo> <mi>Q</mi> <mi mathvariant="normal">Λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AQ=Q\Lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a9b444753de552e5bc83bb9f3d55aeb2d903aeb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.778ex; height:2.509ex;" alt="{\displaystyle AQ=Q\Lambda .}"></dd></dl> Because the columns of Q are linearly independent, Q is invertible. Right multiplying both sides of the equation by Q−1, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=Q\Lambda Q^{-1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mi>Q</mi> <mi mathvariant="normal">Λ</mi> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=Q\Lambda Q^{-1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d116bad0089703e8cb0458ddf2f1b69dfa0c0a1a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.111ex; height:3.009ex;" alt="{\displaystyle A=Q\Lambda Q^{-1},}"></dd></dl> or by instead left multiplying both sides by Q−1, <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Q^{-1}AQ=\Lambda .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>A</mi> <mi>Q</mi> <mo>=</mo> <mi mathvariant="normal">Λ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Q^{-1}AQ=\Lambda .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/415db9aa0f4bb43020fd986d280169052765284c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.111ex; height:3.009ex;" alt="{\displaystyle Q^{-1}AQ=\Lambda .}"></dd></dl> A can therefore be decomposed into a matrix composed of its eigenvectors, a diagonal matrix with its eigenvalues along the diagonal, and the inverse of the matrix of eigenvectors. This is called the <a href="/wiki/Eigendecomposition_of_a_matrix" title="Eigendecomposition of a matrix">eigendecomposition</a> and it is a <a href="/wiki/Matrix_similarity" title="Matrix similarity">similarity transformation</a>. Such a matrix A is said to be similar to the diagonal matrix Λ or <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable</a>. The matrix Q is the change of basis matrix of the similarity transformation. Essentially, the matrices A and Λ represent the same linear transformation expressed in two different bases. The eigenvectors are used as the basis when representing the linear transformation as Λ. Conversely, suppose a matrix A is diagonalizable. Let P be a non-singular square matrix such that P−1AP is some diagonal matrix D. Left multiplying both by P, AP = PD. Each column of P must therefore be an eigenvector of A whose eigenvalue is the corresponding diagonal element of D. Since the columns of P must be linearly independent for P to be invertible, there exist n linearly independent eigenvectors of A. It then follows that the eigenvectors of A form a basis if and only if A is diagonalizable. A matrix that is not diagonalizable is said to be <a href="/wiki/Defective_matrix" title="Defective matrix">defective</a>. For defective matrices, the notion of eigenvectors generalizes to <a href="/wiki/Generalized_eigenvector" title="Generalized eigenvector">generalized eigenvectors</a> and the diagonal matrix of eigenvalues generalizes to the <a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a>. Over an algebraically closed field, any matrix A has a <a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a> and therefore admits a basis of generalized eigenvectors and a decomposition into <a href="/wiki/Generalized_eigenspace" class="mw-redirect" title="Generalized eigenspace">generalized eigenspaces</a>. <div class="mw-heading mw-heading3"><h3 id="Variational_characterization">Variational characterization</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=12" title="Edit section: Variational characterization">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Min-max_theorem" title="Min-max theorem">Min-max theorem</a></div> In the <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a> case, eigenvalues can be given a variational characterization. The largest eigenvalue of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> is the maximum value of the <a href="/wiki/Quadratic_form" title="Quadratic form">quadratic form</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} ^{\textsf {T}}H\mathbf {x} /\mathbf {x} ^{\textsf {T}}\mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} ^{\textsf {T}}H\mathbf {x} /\mathbf {x} ^{\textsf {T}}\mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c719cb0720f9ddaf872405903ba79a3e19cdf4be" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.573ex; height:3.176ex;" alt="{\displaystyle \mathbf {x} ^{\textsf {T}}H\mathbf {x} /\mathbf {x} ^{\textsf {T}}\mathbf {x} }">. A value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {x} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {x} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32adf004df5eb0a8c7fd8c0b6b7405183c5a5ef2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {x} }"> that realizes that maximum is an eigenvector. <div class="mw-heading mw-heading3"><h3 id="Matrix_examples">Matrix examples</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=13" title="Edit section: Matrix examples">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Two-dimensional_matrix_example">Two-dimensional matrix example</h4>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=14" title="Edit section: Two-dimensional matrix example">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Frame"><a href="/wiki/File:Eigenvectors.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/0/06/Eigenvectors.gif" decoding="async" width="300" height="300" class="mw-file-element" data-file-width="300" data-file-height="300" /></a><figcaption>The transformation matrix A = <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{smallmatrix}2&1\\1&2\end{smallmatrix}}\right]}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo>]</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left[{\begin{smallmatrix}2&1\\1&2\end{smallmatrix}}\right]}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/838a30dc9d065ec434dff490bd84061ed569db3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.919ex; margin-bottom: -0.253ex; width:5.108ex; height:3.343ex;" alt="{\displaystyle \left[{\begin{smallmatrix}2&1\\1&2\end{smallmatrix}}\right]}"> preserves the direction of purple vectors parallel to vλ=1 = [1 −1]T and blue vectors parallel to vλ=3 = [1 1]T. The red vectors are not parallel to either eigenvector, so, their directions are changed by the transformation. The lengths of the purple vectors are unchanged after the transformation (due to their eigenvalue of 1), while blue vectors are three times the length of the original (due to their eigenvalue of 3). See also: <a href="/wiki/File:Eigenvectors-extended.gif" title="File:Eigenvectors-extended.gif">An extended version, showing all four quadrants</a>.</figcaption></figure> Consider the matrix <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b50d689a4b5e815e14a9afb3b8da06fd6cc8e723" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:13.342ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}2&1\\1&2\end{bmatrix}}.}"> The figure on the right shows the effect of this transformation on point coordinates in the plane. The eigenvectors v of this transformation satisfy equation (<a href="#math_1">1</a>), and the values of λ for which the determinant of the matrix (A − λI) equals zero are the eigenvalues. Taking the determinant to find characteristic polynomial of A, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&1\\1&2\end{bmatrix}}-\lambda {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}\\[6pt]&=3-4\lambda +\lambda ^{2}\\[6pt]&=(\lambda -3)(\lambda -1).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <mo>−</mo> <mi>λ</mi> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> <mo>−</mo> <mi>λ</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mn>3</mn> <mo>−</mo> <mn>4</mn> <mi>λ</mi> <mo>+</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <mn>3</mn> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <mn>1</mn> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&1\\1&2\end{bmatrix}}-\lambda {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}\\[6pt]&=3-4\lambda +\lambda ^{2}\\[6pt]&=(\lambda -3)(\lambda -1).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/852afe30ae1c99b2f2ff91b62e226d28cef2609a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.005ex; width:55.379ex; height:15.176ex;" alt="{\displaystyle {\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&1\\1&2\end{bmatrix}}-\lambda {\begin{bmatrix}1&0\\0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &1\\1&2-\lambda \end{vmatrix}}\\[6pt]&=3-4\lambda +\lambda ^{2}\\[6pt]&=(\lambda -3)(\lambda -1).\end{aligned}}}"> Setting the characteristic polynomial equal to zero, it has roots at λ=1 and λ=3, which are the two eigenvalues of A. For λ=1, equation (<a href="#math_2">2</a>) becomes, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A-I)\mathbf {v} _{\lambda =1}={\begin{bmatrix}1&1\\1&1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>I</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A-I)\mathbf {v} _{\lambda =1}={\begin{bmatrix}1&1\\1&1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b852b25869b96f6258368c1f7e0e1c312ae607c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:36.075ex; height:6.176ex;" alt="{\displaystyle (A-I)\mathbf {v} _{\lambda =1}={\begin{bmatrix}1&1\\1&1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1v_{1}+1v_{2}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>1</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 1v_{1}+1v_{2}=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/813ba4340c75dd570df5e880d649e6083a937484" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.79ex; height:2.509ex;" alt="{\displaystyle 1v_{1}+1v_{2}=0}"> Any nonzero vector with v1 = −v2 solves this equation. Therefore, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}v_{1}\\-v_{1}\end{bmatrix}}={\begin{bmatrix}1\\-1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}v_{1}\\-v_{1}\end{bmatrix}}={\begin{bmatrix}1\\-1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6b86a8d91f1ccb9b4bcc7089fe30ff61f40281" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:24.272ex; height:6.176ex;" alt="{\displaystyle \mathbf {v} _{\lambda =1}={\begin{bmatrix}v_{1}\\-v_{1}\end{bmatrix}}={\begin{bmatrix}1\\-1\end{bmatrix}}}"> is an eigenvector of A corresponding to λ = 1, as is any scalar multiple of this vector. For λ=3, equation (<a href="#math_2">2</a>) becomes <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}(A-3I)\mathbf {v} _{\lambda =3}&={\begin{bmatrix}-1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}\\-1v_{1}+1v_{2}&=0;\\1v_{1}-1v_{2}&=0\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mn>3</mn> <mi>I</mi> <mo stretchy="false">)</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>=</mo> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <mn>1</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> <mo>;</mo> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>−</mo> <mn>1</mn> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}(A-3I)\mathbf {v} _{\lambda =3}&={\begin{bmatrix}-1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}\\-1v_{1}+1v_{2}&=0;\\1v_{1}-1v_{2}&=0\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63228c273eb8315bc92ee2604596c387d2a2cc7b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:41.605ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}(A-3I)\mathbf {v} _{\lambda =3}&={\begin{bmatrix}-1&1\\1&-1\end{bmatrix}}{\begin{bmatrix}v_{1}\\v_{2}\end{bmatrix}}={\begin{bmatrix}0\\0\end{bmatrix}}\\-1v_{1}+1v_{2}&=0;\\1v_{1}-1v_{2}&=0\end{aligned}}}"> Any nonzero vector with v1 = v2 solves this equation. Therefore, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{\lambda =3}={\begin{bmatrix}v_{1}\\v_{1}\end{bmatrix}}={\begin{bmatrix}1\\1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mo>=</mo> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{\lambda =3}={\begin{bmatrix}v_{1}\\v_{1}\end{bmatrix}}={\begin{bmatrix}1\\1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/722d0163f7f1e8450ccde6ed95fb40a307bed5db" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:20.656ex; height:6.176ex;" alt="{\displaystyle \mathbf {v} _{\lambda =3}={\begin{bmatrix}v_{1}\\v_{1}\end{bmatrix}}={\begin{bmatrix}1\\1\end{bmatrix}}}"> is an eigenvector of A corresponding to λ = 3, as is any scalar multiple of this vector. Thus, the vectors vλ=1 and vλ=3 are eigenvectors of A associated with the eigenvalues λ=1 and λ=3, respectively. <div class="mw-heading mw-heading4"><h4 id="Three-dimensional_matrix_example">Three-dimensional matrix example</h4>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=15" title="Edit section: Three-dimensional matrix example">edit</a>]</div> Consider the matrix <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>9</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f615cf4a3017ca111a13b9bb073d5fcb1136709b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.473ex; height:9.176ex;" alt="{\displaystyle A={\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}.}"> The characteristic polynomial of A is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}-\lambda {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &0&0\\0&3-\lambda &4\\0&4&9-\lambda \end{vmatrix}},\\[6pt]&=(2-\lambda ){\bigl [}(3-\lambda )(9-\lambda )-16{\bigr ]}=-\lambda ^{3}+14\lambda ^{2}-35\lambda +22.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="0.9em 0.3em" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>|</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>9</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>−</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mrow> <mo>|</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <mo>−</mo> <mi>λ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> <mo>−</mo> <mi>λ</mi> </mtd> <mtd> <mn>4</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>4</mn> </mtd> <mtd> <mn>9</mn> <mo>−</mo> <mi>λ</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">[</mo> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>9</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>−</mo> <mn>16</mn> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">]</mo> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mo>+</mo> <mn>14</mn> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>35</mn> <mi>λ</mi> <mo>+</mo> <mn>22.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}-\lambda {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &0&0\\0&3-\lambda &4\\0&4&9-\lambda \end{vmatrix}},\\[6pt]&=(2-\lambda ){\bigl [}(3-\lambda )(9-\lambda )-16{\bigr ]}=-\lambda ^{3}+14\lambda ^{2}-35\lambda +22.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdfec3c58ac4306d8cc19110ac4b2b5bfbea234e" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.505ex; width:71.968ex; height:14.176ex;" alt="{\displaystyle {\begin{aligned}\det(A-\lambda I)&=\left|{\begin{bmatrix}2&0&0\\0&3&4\\0&4&9\end{bmatrix}}-\lambda {\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}}\right|={\begin{vmatrix}2-\lambda &0&0\\0&3-\lambda &4\\0&4&9-\lambda \end{vmatrix}},\\[6pt]&=(2-\lambda ){\bigl [}(3-\lambda )(9-\lambda )-16{\bigr ]}=-\lambda ^{3}+14\lambda ^{2}-35\lambda +22.\end{aligned}}}"> The roots of the characteristic polynomial are 2, 1, and 11, which are the only three eigenvalues of A. These eigenvalues correspond to the eigenvectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&0&0\end{bmatrix}}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&0&0\end{bmatrix}}^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c98076f37531d66e12a3985aa85473096e1ace07" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.529ex; height:3.343ex;" alt="{\displaystyle {\begin{bmatrix}1&0&0\end{bmatrix}}^{\textsf {T}}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&-2&1\end{bmatrix}}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>2</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&-2&1\end{bmatrix}}^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35e079651d4cd169776544ffe097b84282c996aa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.337ex; height:3.343ex;" alt="{\displaystyle {\begin{bmatrix}0&-2&1\end{bmatrix}}^{\textsf {T}}}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&1&2\end{bmatrix}}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&1&2\end{bmatrix}}^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c70f4e766a0f65f0c8b4eab9683ce91e85c1254" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.529ex; height:3.343ex;" alt="{\displaystyle {\begin{bmatrix}0&1&2\end{bmatrix}}^{\textsf {T}}}">, or any nonzero multiple thereof. <div class="mw-heading mw-heading4"><h4 id="Three-dimensional_matrix_example_with_complex_eigenvalues">Three-dimensional matrix example with complex eigenvalues</h4>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=16" title="Edit section: Three-dimensional matrix example with complex eigenvalues">edit</a>]</div> Consider the <a href="/wiki/Permutation_matrix" title="Permutation matrix">cyclic permutation matrix</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d07fa59ff389f7f38682c5a7dfb7edd578aa238" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.473ex; height:9.176ex;" alt="{\displaystyle A={\begin{bmatrix}0&1&0\\0&0&1\\1&0&0\end{bmatrix}}.}"> This matrix shifts the coordinates of the vector up by one position and moves the first coordinate to the bottom. Its characteristic polynomial is 1 − λ3, whose roots are <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\lambda _{1}&=1\\\lambda _{2}&=-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}\\\lambda _{3}&=\lambda _{2}^{*}=-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>+</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>=</mo> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>−</mo> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msqrt> <mn>3</mn> </msqrt> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\lambda _{1}&=1\\\lambda _{2}&=-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}\\\lambda _{3}&=\lambda _{2}^{*}=-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dfdb45f490f28d61f12ff91e08e4b11a789b7fe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:23.151ex; height:14.843ex;" alt="{\displaystyle {\begin{aligned}\lambda _{1}&=1\\\lambda _{2}&=-{\frac {1}{2}}+i{\frac {\sqrt {3}}{2}}\\\lambda _{3}&=\lambda _{2}^{*}=-{\frac {1}{2}}-i{\frac {\sqrt {3}}{2}}\end{aligned}}}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}"> is an <a href="/wiki/Imaginary_unit" title="Imaginary unit">imaginary unit</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i^{2}=-1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>i</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mo>−</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i^{2}=-1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/88e98a401d352e5037d5043028e2d7f449e83fa6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.926ex; height:2.843ex;" alt="{\displaystyle i^{2}=-1}">. For the real eigenvalue λ1 = 1, any vector with three equal nonzero entries is an eigenvector. For example, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\begin{bmatrix}5\\5\\5\end{bmatrix}}={\begin{bmatrix}5\\5\\5\end{bmatrix}}=1\cdot {\begin{bmatrix}5\\5\\5\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mn>1</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> <mtr> <mtd> <mn>5</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\begin{bmatrix}5\\5\\5\end{bmatrix}}={\begin{bmatrix}5\\5\\5\end{bmatrix}}=1\cdot {\begin{bmatrix}5\\5\\5\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/04948713f37cdde1ecc99e80d033d6b6640d288b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:26.472ex; height:9.176ex;" alt="{\displaystyle A{\begin{bmatrix}5\\5\\5\end{bmatrix}}={\begin{bmatrix}5\\5\\5\end{bmatrix}}=1\cdot {\begin{bmatrix}5\\5\\5\end{bmatrix}}.}"> For the complex conjugate pair of imaginary eigenvalues, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{2}\lambda _{3}=1,\quad \lambda _{2}^{2}=\lambda _{3},\quad \lambda _{3}^{2}=\lambda _{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>,</mo> <mspace width="1em" /> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{2}\lambda _{3}=1,\quad \lambda _{2}^{2}=\lambda _{3},\quad \lambda _{3}^{2}=\lambda _{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4fbd4510cf1a9c152ec8d6db6ffa561bb1c309c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:32.275ex; height:3.176ex;" alt="{\displaystyle \lambda _{2}\lambda _{3}=1,\quad \lambda _{2}^{2}=\lambda _{3},\quad \lambda _{3}^{2}=\lambda _{2}.}"> Then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}}={\begin{bmatrix}\lambda _{2}\\\lambda _{3}\\1\end{bmatrix}}=\lambda _{2}\cdot {\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}}={\begin{bmatrix}\lambda _{2}\\\lambda _{3}\\1\end{bmatrix}}=\lambda _{2}\cdot {\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/74ab7f86dd4cd11bd0b3bb0a8349b445285342f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:31.46ex; height:9.176ex;" alt="{\displaystyle A{\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}}={\begin{bmatrix}\lambda _{2}\\\lambda _{3}\\1\end{bmatrix}}=\lambda _{2}\cdot {\begin{bmatrix}1\\\lambda _{2}\\\lambda _{3}\end{bmatrix}},}"> and <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A{\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}={\begin{bmatrix}\lambda _{3}\\\lambda _{2}\\1\end{bmatrix}}=\lambda _{3}\cdot {\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A{\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}={\begin{bmatrix}\lambda _{3}\\\lambda _{2}\\1\end{bmatrix}}=\lambda _{3}\cdot {\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40a43c18286d415767f6573e76da6fa4da9a3ba6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:31.46ex; height:9.176ex;" alt="{\displaystyle A{\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}={\begin{bmatrix}\lambda _{3}\\\lambda _{2}\\1\end{bmatrix}}=\lambda _{3}\cdot {\begin{bmatrix}1\\\lambda _{3}\\\lambda _{2}\end{bmatrix}}.}"> Therefore, the other two eigenvectors of A are complex and are <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}1&\lambda _{2}&\lambda _{3}\end{bmatrix}}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}1&\lambda _{2}&\lambda _{3}\end{bmatrix}}^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a90781b4c41d6ff5279befcc0713c266f1bdf552" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.555ex; height:3.509ex;" alt="{\displaystyle \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}1&\lambda _{2}&\lambda _{3}\end{bmatrix}}^{\textsf {T}}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}1&\lambda _{3}&\lambda _{2}\end{bmatrix}}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mtd> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}1&\lambda _{3}&\lambda _{2}\end{bmatrix}}^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a6ac598b971356daf89b9188a523c9fe27748356" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.555ex; height:3.509ex;" alt="{\displaystyle \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}1&\lambda _{3}&\lambda _{2}\end{bmatrix}}^{\textsf {T}}}"> with eigenvalues λ2 and λ3, respectively. The two complex eigenvectors also appear in a complex conjugate pair, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{\lambda _{2}}=\mathbf {v} _{\lambda _{3}}^{*}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{\lambda _{2}}=\mathbf {v} _{\lambda _{3}}^{*}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9af7bb9f70791f420f0e216a7979da47a372c4c1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:10.611ex; height:3.176ex;" alt="{\displaystyle \mathbf {v} _{\lambda _{2}}=\mathbf {v} _{\lambda _{3}}^{*}.}"> <div class="mw-heading mw-heading4"><h4 id="Diagonal_matrix_example">Diagonal matrix example</h4>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=17" title="Edit section: Diagonal matrix example">edit</a>]</div> Matrices with entries only along the main diagonal are called <a href="/wiki/Diagonal_matrices" class="mw-redirect" title="Diagonal matrices">diagonal matrices</a>. The eigenvalues of a diagonal matrix are the diagonal elements themselves. Consider the matrix <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a15745a43931c2813fc4db090c18d4000f48ae6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.473ex; height:9.176ex;" alt="{\displaystyle A={\begin{bmatrix}1&0&0\\0&2&0\\0&0&3\end{bmatrix}}.}"> The characteristic polynomial of A is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53b873636a4c497afad1d648cdc49fcb253c61a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.397ex; height:2.843ex;" alt="{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}"> which has the roots λ1 = 1, λ2 = 2, and λ3 = 3. These roots are the diagonal elements as well as the eigenvalues of A. Each diagonal element corresponds to an eigenvector whose only nonzero component is in the same row as that diagonal element. In the example, the eigenvalues correspond to the eigenvectors, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{\lambda _{1}}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}0\\1\\0\end{bmatrix}},\quad \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{\lambda _{1}}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}0\\1\\0\end{bmatrix}},\quad \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/517b833a3a023e44fa72a12778b1ad4d5d1a695a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:41.998ex; height:9.176ex;" alt="{\displaystyle \mathbf {v} _{\lambda _{1}}={\begin{bmatrix}1\\0\\0\end{bmatrix}},\quad \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}0\\1\\0\end{bmatrix}},\quad \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}"> respectively, as well as scalar multiples of these vectors. <div class="mw-heading mw-heading4"><h4 id="Triangular_matrix_example">Triangular matrix example</h4>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=18" title="Edit section: Triangular matrix example">edit</a>]</div> A matrix whose elements above the main diagonal are all zero is called a lower <a href="/wiki/Triangular_matrix" title="Triangular matrix">triangular matrix</a>, while a matrix whose elements below the main diagonal are all zero is called an upper triangular matrix. As with diagonal matrices, the eigenvalues of triangular matrices are the elements of the main diagonal. Consider the lower triangular matrix, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}1&0&0\\1&2&0\\2&3&3\end{bmatrix}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}1&0&0\\1&2&0\\2&3&3\end{bmatrix}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94001b6bc1df289d4d96359eeb936b17e67957cd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:17.473ex; height:9.176ex;" alt="{\displaystyle A={\begin{bmatrix}1&0&0\\1&2&0\\2&3&3\end{bmatrix}}.}"> The characteristic polynomial of A is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mn>1</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mi>λ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53b873636a4c497afad1d648cdc49fcb253c61a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.397ex; height:2.843ex;" alt="{\displaystyle \det(A-\lambda I)=(1-\lambda )(2-\lambda )(3-\lambda ),}"> which has the roots λ1 = 1, λ2 = 2, and λ3 = 3. These roots are the diagonal elements as well as the eigenvalues of A. These eigenvalues correspond to the eigenvectors, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{\lambda _{1}}={\begin{bmatrix}1\\-1\\{\frac {1}{2}}\end{bmatrix}},\quad \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}0\\1\\-3\end{bmatrix}},\quad \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> <mspace width="1em" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{\lambda _{1}}={\begin{bmatrix}1\\-1\\{\frac {1}{2}}\end{bmatrix}},\quad \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}0\\1\\-3\end{bmatrix}},\quad \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dfb33470ae800e4881edb852c6c115b029cd6a57" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.505ex; width:45.615ex; height:10.176ex;" alt="{\displaystyle \mathbf {v} _{\lambda _{1}}={\begin{bmatrix}1\\-1\\{\frac {1}{2}}\end{bmatrix}},\quad \mathbf {v} _{\lambda _{2}}={\begin{bmatrix}0\\1\\-3\end{bmatrix}},\quad \mathbf {v} _{\lambda _{3}}={\begin{bmatrix}0\\0\\1\end{bmatrix}},}"> respectively, as well as scalar multiples of these vectors. <div class="mw-heading mw-heading4"><h4 id="Matrix_with_repeated_eigenvalues_example">Matrix with repeated eigenvalues example</h4>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=19" title="Edit section: Matrix with repeated eigenvalues example">edit</a>]</div> As in the previous example, the lower triangular matrix <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}2&0&0&0\\1&2&0&0\\0&1&3&0\\0&0&1&3\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}2&0&0&0\\1&2&0&0\\0&1&3&0\\0&0&1&3\end{bmatrix}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0326a200a7af43eb04857083054bb9d7429064ca" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:20.958ex; height:12.509ex;" alt="{\displaystyle A={\begin{bmatrix}2&0&0&0\\1&2&0&0\\0&1&3&0\\0&0&1&3\end{bmatrix}},}"> has a characteristic polynomial that is the product of its diagonal elements, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &0&0&0\\1&2-\lambda &0&0\\0&1&3-\lambda &0\\0&0&1&3-\lambda \end{vmatrix}}=(2-\lambda )^{2}(3-\lambda )^{2}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>λ</mi> <mi>I</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>|</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>2</mn> <mo>−</mo> <mi>λ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>2</mn> <mo>−</mo> <mi>λ</mi> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> <mo>−</mo> <mi>λ</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>3</mn> <mo>−</mo> <mi>λ</mi> </mtd> </mtr> </mtable> <mo>|</mo> </mrow> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <mn>2</mn> <mo>−</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo stretchy="false">(</mo> <mn>3</mn> <mo>−</mo> <mi>λ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &0&0&0\\1&2-\lambda &0&0\\0&1&3-\lambda &0\\0&0&1&3-\lambda \end{vmatrix}}=(2-\lambda )^{2}(3-\lambda )^{2}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b7f319c2059f56f5af3fc4c2df386e814a12731" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:65.882ex; height:12.509ex;" alt="{\displaystyle \det(A-\lambda I)={\begin{vmatrix}2-\lambda &0&0&0\\1&2-\lambda &0&0\\0&1&3-\lambda &0\\0&0&1&3-\lambda \end{vmatrix}}=(2-\lambda )^{2}(3-\lambda )^{2}.}"> The roots of this polynomial, and hence the eigenvalues, are 2 and 3. The algebraic multiplicity of each eigenvalue is 2; in other words they are both double roots. The sum of the algebraic multiplicities of all distinct eigenvalues is μA = 4 = n, the order of the characteristic polynomial and the dimension of A. On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&1&-1&1\end{bmatrix}}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&1&-1&1\end{bmatrix}}^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/980f42653cb2ddd817440f41a74722af04ad721c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.822ex; height:3.343ex;" alt="{\displaystyle {\begin{bmatrix}0&1&-1&1\end{bmatrix}}^{\textsf {T}}}"> and is therefore 1-dimensional. Similarly, the geometric multiplicity of the eigenvalue 3 is 1 because its eigenspace is spanned by just one vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}0&0&0&1\end{bmatrix}}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}0&0&0&1\end{bmatrix}}^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11704eedc136af15883e9ee6370c77a6f03dcf5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.014ex; height:3.343ex;" alt="{\displaystyle {\begin{bmatrix}0&0&0&1\end{bmatrix}}^{\textsf {T}}}">. The total geometric multiplicity γA is 2, which is the smallest it could be for a matrix with two distinct eigenvalues. Geometric multiplicities are defined in a later section. <div class="mw-heading mw-heading3"><h3 id="Eigenvector-eigenvalue_identity">Eigenvector-eigenvalue identity</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=20" title="Edit section: Eigenvector-eigenvalue identity">edit</a>]</div> For a <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian matrix</a>, the norm squared of the jth component of a normalized eigenvector can be calculated using only the matrix eigenvalues and the eigenvalues of the corresponding <a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">minor matrix</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |v_{i,j}|^{2}={\frac {\prod _{k}{(\lambda _{i}-\lambda _{k}(M_{j}))}}{\prod _{k\neq i}{(\lambda _{i}-\lambda _{k})}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> </mrow> <mrow> <munder> <mo>∏</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>≠</mo> <mi>i</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>−</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |v_{i,j}|^{2}={\frac {\prod _{k}{(\lambda _{i}-\lambda _{k}(M_{j}))}}{\prod _{k\neq i}{(\lambda _{i}-\lambda _{k})}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/abd699bec7ae8216e7baaf247e937d3837aeb469" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:27.883ex; height:6.843ex;" alt="{\displaystyle |v_{i,j}|^{2}={\frac {\prod _{k}{(\lambda _{i}-\lambda _{k}(M_{j}))}}{\prod _{k\neq i}{(\lambda _{i}-\lambda _{k})}}},}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle M_{j}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <msub> <mi>M</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle M_{j}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e533cedf4b286ac7c493ac24d10ae582ca754ea4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.164ex; height:2.843ex;" alt="{\textstyle M_{j}}"> is the <a href="/wiki/Submatrix" class="mw-redirect" title="Submatrix">submatrix</a> formed by removing the jth row and column from the original matrix.<a href="#cite_note-FOOTNOTEWolchover2019-36">[33]</a><a href="#cite_note-FOOTNOTEDentonParkeTaoZhang2022-37">[34]</a><a href="#cite_note-FOOTNOTEVan_Mieghem2014-38">[35]</a> This identity also extends to <a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">diagonalizable matrices</a>, and has been rediscovered many times in the literature.<a href="#cite_note-FOOTNOTEDentonParkeTaoZhang2022-37">[34]</a><a href="#cite_note-FOOTNOTEVan_Mieghem2024-39">[36]</a> <div class="mw-heading mw-heading2"><h2 id="Eigenvalues_and_eigenfunctions_of_differential_operators">Eigenvalues and eigenfunctions of differential operators</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=21" title="Edit section: Eigenvalues and eigenfunctions of differential operators">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Eigenfunction" title="Eigenfunction">Eigenfunction</a></div> The definitions of eigenvalue and eigenvectors of a linear transformation T remains valid even if the underlying vector space is an infinite-dimensional <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert</a> or <a href="/wiki/Banach_space" title="Banach space">Banach space</a>. A widely used class of linear transformations acting on infinite-dimensional spaces are the <a href="/wiki/Differential_operator" title="Differential operator">differential operators</a> on <a href="/wiki/Function_space" title="Function space">function spaces</a>. Let D be a linear differential operator on the space C∞ of infinitely <a href="/wiki/Derivative" title="Derivative">differentiable</a> real functions of a real argument t. The eigenvalue equation for D is the <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Df(t)=\lambda f(t)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Df(t)=\lambda f(t)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17eeb61dc592bfd36df5a7e5fd733480d3110e1d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.233ex; height:2.843ex;" alt="{\displaystyle Df(t)=\lambda f(t)}"> The functions that satisfy this equation are eigenvectors of D and are commonly called eigenfunctions. <div class="mw-heading mw-heading3"><h3 id="Derivative_operator_example">Derivative operator example</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=22" title="Edit section: Derivative operator example">edit</a>]</div> Consider the derivative operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {d}{dt}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\tfrac {d}{dt}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/db1b2f034e8a0911ef7410d16ca3402e31f76d9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.29ex; height:3.843ex;" alt="{\displaystyle {\tfrac {d}{dt}}}"> with eigenvalue equation <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d}{dt}}f(t)=\lambda f(t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>d</mi> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d}{dt}}f(t)=\lambda f(t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/57b0e4ed327d8b51b8bb142e8c31f27e118a64e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.847ex; height:5.509ex;" alt="{\displaystyle {\frac {d}{dt}}f(t)=\lambda f(t).}"> This differential equation can be solved by multiplying both sides by dt/f(t) and <a href="/wiki/Integration_(calculus)" class="mw-redirect" title="Integration (calculus)">integrating</a>. Its solution, the <a href="/wiki/Exponential_function" title="Exponential function">exponential function</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t)=f(0)e^{\lambda t},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>f</mi> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>λ</mi> <mi>t</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(t)=f(0)e^{\lambda t},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af76facaf4f0468b8be2b6673e3e2703f9e14adf" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.791ex; height:3.176ex;" alt="{\displaystyle f(t)=f(0)e^{\lambda t},}"> is the eigenfunction of the derivative operator. In this case the eigenfunction is itself a function of its associated eigenvalue. In particular, for λ = 0 the eigenfunction f(t) is a constant. The main <a href="/wiki/Eigenfunction" title="Eigenfunction">eigenfunction</a> article gives other examples. <div class="mw-heading mw-heading2"><h2 id="General_definition">General definition</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=23" title="Edit section: General definition">edit</a>]</div> The concept of eigenvalues and eigenvectors extends naturally to arbitrary <a href="/wiki/Linear_map" title="Linear map">linear transformations</a> on arbitrary vector spaces. Let V be any vector space over some <a href="/wiki/Field_(algebra)" class="mw-redirect" title="Field (algebra)">field</a> K of <a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">scalars</a>, and let T be a linear transformation mapping V into V, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T:V\to V.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo>:</mo> <mi>V</mi> <mo stretchy="false">→</mo> <mi>V</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T:V\to V.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0cddc5c4616e0e0636cdbe8d5aab54c07627f231" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.409ex; height:2.176ex;" alt="{\displaystyle T:V\to V.}"> We say that a nonzero vector v ∈ V is an eigenvector of T if and only if there exists a scalar λ ∈ K such that <table role="presentation" style="border-collapse:collapse; margin:0 0 0 1.6em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/668a03ab032405621214d60ee473182f9288b771" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.368ex; height:2.843ex;" alt="{\displaystyle T(\mathbf {v} )=\lambda \mathbf {v} .}"> </td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(5)</td></tr></tbody></table> This equation is called the eigenvalue equation for T, and the scalar λ is the eigenvalue of T corresponding to the eigenvector v. T(v) is the result of applying the transformation T to the vector v, while λv is the product of the scalar λ with v.<a href="#cite_note-FOOTNOTEKornKorn2000Section_14.3.5a-40">[37]</a><a href="#cite_note-FOOTNOTEFriedbergInselSpence1989p._217-41">[38]</a> <div class="mw-heading mw-heading3"><h3 id="Eigenspaces,_geometric_multiplicity,_and_the_eigenbasis">Eigenspaces, geometric multiplicity, and the eigenbasis</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=24" title="Edit section: Eigenspaces, geometric multiplicity, and the eigenbasis">edit</a>]</div> Given an eigenvalue λ, consider the set <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E=\left\{\mathbf {v} :T(\mathbf {v} )=\lambda \mathbf {v} \right\},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> <mo>=</mo> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo>:</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mo>}</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E=\left\{\mathbf {v} :T(\mathbf {v} )=\lambda \mathbf {v} \right\},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dce0fc9ef64d044ad0de42017896de6d64ef5d90" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.302ex; height:2.843ex;" alt="{\displaystyle E=\left\{\mathbf {v} :T(\mathbf {v} )=\lambda \mathbf {v} \right\},}"> which is the union of the zero vector with the set of all eigenvectors associated with λ. E is called the eigenspace or characteristic space of T associated with λ.<a href="#cite_note-42">[39]</a> By definition of a linear transformation, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}T(\mathbf {x} +\mathbf {y} )&=T(\mathbf {x} )+T(\mathbf {y} ),\\T(\alpha \mathbf {x} )&=\alpha T(\mathbf {x} ),\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>+</mo> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">y</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>α</mi> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">x</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}T(\mathbf {x} +\mathbf {y} )&=T(\mathbf {x} )+T(\mathbf {y} ),\\T(\alpha \mathbf {x} )&=\alpha T(\mathbf {x} ),\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9a7a7f284c05b6c7c339ddf47ceef2860336d29b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:26.158ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}T(\mathbf {x} +\mathbf {y} )&=T(\mathbf {x} )+T(\mathbf {y} ),\\T(\alpha \mathbf {x} )&=\alpha T(\mathbf {x} ),\end{aligned}}}"> for x, y ∈ V and α ∈ K. Therefore, if u and v are eigenvectors of T associated with eigenvalue λ, namely u, v ∈ E, then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}T(\mathbf {u} +\mathbf {v} )&=\lambda (\mathbf {u} +\mathbf {v} ),\\T(\alpha \mathbf {v} )&=\lambda (\alpha \mathbf {v} ).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>T</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>λ</mi> <mo stretchy="false">(</mo> <mi>α</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mo stretchy="false">)</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}T(\mathbf {u} +\mathbf {v} )&=\lambda (\mathbf {u} +\mathbf {v} ),\\T(\alpha \mathbf {v} )&=\lambda (\alpha \mathbf {v} ).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/68db77cf97dffdf2e6aa3fcce27465fb407f8001" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.58ex; height:6.176ex;" alt="{\displaystyle {\begin{aligned}T(\mathbf {u} +\mathbf {v} )&=\lambda (\mathbf {u} +\mathbf {v} ),\\T(\alpha \mathbf {v} )&=\lambda (\alpha \mathbf {v} ).\end{aligned}}}"> So, both u + v and αv are either zero or eigenvectors of T associated with λ, namely u + v, αv ∈ E, and E is closed under addition and scalar multiplication. The eigenspace E associated with λ is therefore a linear subspace of V.<a href="#cite_note-43">[40]</a> If that subspace has dimension 1, it is sometimes called an eigenline.<a href="#cite_note-FOOTNOTELipschutzLipson2002111-44">[41]</a> The geometric multiplicity γT(λ) of an eigenvalue λ is the dimension of the eigenspace associated with λ, i.e., the maximum number of linearly independent eigenvectors associated with that eigenvalue.<a href="#cite_note-FOOTNOTENering1970107-9">[9]</a><a href="#cite_note-FOOTNOTEGolubVan_Loan1996316-29">[26]</a><a href="#cite_note-FOOTNOTERoman2008p._189_§8-45">[42]</a> By the definition of eigenvalues and eigenvectors, γT(λ) ≥ 1 because every eigenvalue has at least one eigenvector. The eigenspaces of T always form a <a href="/wiki/Direct_sum" title="Direct sum">direct sum</a>. As a consequence, eigenvectors of different eigenvalues are always linearly independent. Therefore, the sum of the dimensions of the eigenspaces cannot exceed the dimension n of the vector space on which T operates, and there cannot be more than n distinct eigenvalues.<a href="#cite_note-46">[d]</a> Any subspace spanned by eigenvectors of T is an <a href="/wiki/Invariant_subspace" title="Invariant subspace">invariant subspace</a> of T, and the restriction of T to such a subspace is diagonalizable. Moreover, if the entire vector space V can be spanned by the eigenvectors of T, or equivalently if the direct sum of the eigenspaces associated with all the eigenvalues of T is the entire vector space V, then a basis of V called an eigenbasis can be formed from linearly independent eigenvectors of T. When T admits an eigenbasis, T is diagonalizable. <div class="mw-heading mw-heading3"><h3 id="Spectral_theory">Spectral theory</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=25" title="Edit section: Spectral theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Spectral_theory" title="Spectral theory">Spectral theory</a></div> If λ is an eigenvalue of T, then the operator (T − λI) is not <a href="/wiki/One_to_one_correspondence" class="mw-redirect" title="One to one correspondence">one-to-one</a>, and therefore its inverse (T − λI)−1 does not exist. The converse is true for finite-dimensional vector spaces, but not for infinite-dimensional vector spaces. In general, the operator (T − λI) may not have an inverse even if λ is not an eigenvalue. For this reason, in <a href="/wiki/Functional_analysis" title="Functional analysis">functional analysis</a> eigenvalues can be generalized to the <a href="/wiki/Spectrum_(functional_analysis)" title="Spectrum (functional analysis)">spectrum of a linear operator</a> T as the set of all scalars λ for which the operator (T − λI) has no <a href="/wiki/Bounded_operator" title="Bounded operator">bounded</a> inverse. The spectrum of an operator always contains all its eigenvalues but is not limited to them. <div class="mw-heading mw-heading3"><h3 id="Associative_algebras_and_representation_theory">Associative algebras and representation theory</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=26" title="Edit section: Associative algebras and representation theory">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Weight_(representation_theory)" title="Weight (representation theory)">Weight (representation theory)</a></div> One can generalize the algebraic object that is acting on the vector space, replacing a single operator acting on a vector space with an <a href="/wiki/Algebra_representation" title="Algebra representation">algebra representation</a> – an <a href="/wiki/Associative_algebra" title="Associative algebra">associative algebra</a> acting on a <a href="/wiki/Module_(mathematics)" title="Module (mathematics)">module</a>. The study of such actions is the field of <a href="/wiki/Representation_theory" title="Representation theory">representation theory</a>. The <a href="/wiki/Weight_(representation_theory)" title="Weight (representation theory)">representation-theoretical concept of weight</a> is an analog of eigenvalues, while weight vectors and weight spaces are the analogs of eigenvectors and eigenspaces, respectively. <a href="/wiki/Hecke_eigensheaf" title="Hecke eigensheaf">Hecke eigensheaf</a> is a tensor-multiple of itself and is considered in <a href="/wiki/Langlands_correspondence" class="mw-redirect" title="Langlands correspondence">Langlands correspondence</a>. <div class="mw-heading mw-heading2"><h2 id="Dynamic_equations">Dynamic equations</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=27" title="Edit section: Dynamic equations">edit</a>]</div> The simplest <a href="/wiki/Difference_equation" class="mw-redirect" title="Difference equation">difference equations</a> have the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots +a_{k}x_{t-k}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mi>k</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots +a_{k}x_{t-k}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20f93d6d1cb5ea5a81cbb401b0c0c64f26640ab0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:36.835ex; height:2.343ex;" alt="{\displaystyle x_{t}=a_{1}x_{t-1}+a_{2}x_{t-2}+\cdots +a_{k}x_{t-k}.}"></dd></dl> The solution of this equation for x in terms of t is found by using its characteristic equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{k}-a_{1}\lambda ^{k-1}-a_{2}\lambda ^{k-2}-\cdots -a_{k-1}\lambda -a_{k}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mo>⋯</mo> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mi>λ</mi> <mo>−</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{k}-a_{1}\lambda ^{k-1}-a_{2}\lambda ^{k-2}-\cdots -a_{k-1}\lambda -a_{k}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6515ebb59338c27a42b93bd4d1d3cfc0dc23f3e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:46.027ex; height:3.009ex;" alt="{\displaystyle \lambda ^{k}-a_{1}\lambda ^{k-1}-a_{2}\lambda ^{k-2}-\cdots -a_{k-1}\lambda -a_{k}=0,}"></dd></dl> which can be found by stacking into matrix form a set of equations consisting of the above difference equation and the k – 1 equations <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t-1}=x_{t-1},\ \dots ,\ x_{t-k+1}=x_{t-k+1},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mtext> </mtext> <mo>…</mo> <mo>,</mo> <mtext> </mtext> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{t-1}=x_{t-1},\ \dots ,\ x_{t-k+1}=x_{t-k+1},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1b557501f46c3d1babde84e9ad23af687950e72" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:34.865ex; height:2.009ex;" alt="{\displaystyle x_{t-1}=x_{t-1},\ \dots ,\ x_{t-k+1}=x_{t-k+1},}"> giving a k-dimensional system of the first order in the stacked variable vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}x_{t}&\cdots &x_{t-k+1}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> <mo>−</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}x_{t}&\cdots &x_{t-k+1}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2c2665afe9ba81713460b81d80ac6f5976244d25" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.591ex; margin-bottom: -0.247ex; width:17.96ex; height:2.843ex;" alt="{\displaystyle {\begin{bmatrix}x_{t}&\cdots &x_{t-k+1}\end{bmatrix}}}"> in terms of its once-lagged value, and taking the characteristic equation of this system's matrix. This equation gives k characteristic roots <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1},\,\ldots ,\,\lambda _{k},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> <mo>,</mo> <mspace width="thinmathspace" /> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1},\,\ldots ,\,\lambda _{k},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca4446ea99db55253a05562c91c1ddf82f4e2352" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.453ex; height:2.509ex;" alt="{\displaystyle \lambda _{1},\,\ldots ,\,\lambda _{k},}"> for use in the solution equation <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{k}\lambda _{k}^{t}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msubsup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msubsup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{k}\lambda _{k}^{t}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e6238c9289d8f2d6ffa8488256956fe09190fd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:23.315ex; height:3.176ex;" alt="{\displaystyle x_{t}=c_{1}\lambda _{1}^{t}+\cdots +c_{k}\lambda _{k}^{t}.}"></dd></dl> A similar procedure is used for solving a <a href="/wiki/Differential_equation" title="Differential equation">differential equation</a> of the form <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {d^{k}x}{dt^{k}}}+a_{k-1}{\frac {d^{k-1}x}{dt^{k-1}}}+\cdots +a_{1}{\frac {dx}{dt}}+a_{0}x=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi>d</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mo>⋯</mo> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>d</mi> <mi>x</mi> </mrow> <mrow> <mi>d</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mi>x</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {d^{k}x}{dt^{k}}}+a_{k-1}{\frac {d^{k-1}x}{dt^{k-1}}}+\cdots +a_{1}{\frac {dx}{dt}}+a_{0}x=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/648a4571806d7fb42eda2abb97c324dfda2ca643" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:43.737ex; height:6.009ex;" alt="{\displaystyle {\frac {d^{k}x}{dt^{k}}}+a_{k-1}{\frac {d^{k-1}x}{dt^{k-1}}}+\cdots +a_{1}{\frac {dx}{dt}}+a_{0}x=0.}"></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Calculation">Calculation</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=28" title="Edit section: Calculation">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Eigenvalue_algorithm" title="Eigenvalue algorithm">Eigenvalue algorithm</a></div> The calculation of eigenvalues and eigenvectors is a topic where theory, as presented in elementary linear algebra textbooks, is often very far from practice. <div class="mw-heading mw-heading3"><h3 id="Classical_method">Classical method</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=29" title="Edit section: Classical method">edit</a>]</div> The classical method is to first find the eigenvalues, and then calculate the eigenvectors for each eigenvalue. It is in several ways poorly suited for non-exact arithmetics such as <a href="/wiki/Floating-point" class="mw-redirect" title="Floating-point">floating-point</a>. <div class="mw-heading mw-heading4"><h4 id="Eigenvalues">Eigenvalues</h4>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=30" title="Edit section: Eigenvalues">edit</a>]</div> The eigenvalues of a matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> can be determined by finding the roots of the characteristic polynomial. This is easy for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 2\times 2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>2</mn> <mo>×</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2\times 2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a0e3400ffb97d67c00267ed50cddfe824cbe80" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.165ex; height:2.176ex;" alt="{\displaystyle 2\times 2}"> matrices, but the difficulty increases rapidly with the size of the matrix. In theory, the coefficients of the characteristic polynomial can be computed exactly, since they are sums of products of matrix elements; and there are algorithms that can find all the roots of a polynomial of arbitrary degree to any required <a href="/wiki/Accuracy" class="mw-redirect" title="Accuracy">accuracy</a>.<a href="#cite_note-FOOTNOTETrefethenBau1997-47">[43]</a> However, this approach is not viable in practice because the coefficients would be contaminated by unavoidable <a href="/wiki/Round-off_error" title="Round-off error">round-off errors</a>, and the roots of a polynomial can be an extremely sensitive function of the coefficients (as exemplified by <a href="/wiki/Wilkinson%27s_polynomial" title="Wilkinson's polynomial">Wilkinson's polynomial</a>).<a href="#cite_note-FOOTNOTETrefethenBau1997-47">[43]</a> Even for matrices whose elements are integers the calculation becomes nontrivial, because the sums are very long; the constant term is the <a href="/wiki/Determinant" title="Determinant">determinant</a>, which for an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> matrix is a sum of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n!}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>!</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n!}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bae971720be3cc9b8d82f4cdac89cb89877514a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.042ex; height:2.176ex;" alt="{\displaystyle n!}"> different products.<a href="#cite_note-48">[e]</a> Explicit <a href="/wiki/Algebraic_solution" class="mw-redirect" title="Algebraic solution">algebraic formulas</a> for the roots of a polynomial exist only if the degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is 4 or less. According to the <a href="/wiki/Abel%E2%80%93Ruffini_theorem" title="Abel–Ruffini theorem">Abel–Ruffini theorem</a> there is no general, explicit and exact algebraic formula for the roots of a polynomial with degree 5 or more. (Generality matters because any polynomial with degree <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> is the characteristic polynomial of some <a href="/wiki/Companion_matrix" title="Companion matrix">companion matrix</a> of order <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}">.) Therefore, for matrices of order 5 or more, the eigenvalues and eigenvectors cannot be obtained by an explicit algebraic formula, and must therefore be computed by approximate <a href="/wiki/Numerical_method" title="Numerical method">numerical methods</a>. Even the <a href="/wiki/Cubic_function#General_solution_to_the_cubic_equation_with_real_coefficients" title="Cubic function">exact formula</a> for the roots of a degree 3 polynomial is numerically impractical. <div class="mw-heading mw-heading4"><h4 id="Eigenvectors">Eigenvectors</h4>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=31" title="Edit section: Eigenvectors">edit</a>]</div> Once the (exact) value of an eigenvalue is known, the corresponding eigenvectors can be found by finding nonzero solutions of the eigenvalue equation, that becomes a <a href="/wiki/Linear_system" title="Linear system">system of linear equations</a> with known coefficients. For example, once it is known that 6 is an eigenvalue of the matrix <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\begin{bmatrix}4&1\\6&3\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\begin{bmatrix}4&1\\6&3\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eab7c7f2958fa6428a7556c1747ec0ae030454c5" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:12.696ex; height:6.176ex;" alt="{\displaystyle A={\begin{bmatrix}4&1\\6&3\end{bmatrix}}}"> we can find its eigenvectors by solving the equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Av=6v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mi>v</mi> <mo>=</mo> <mn>6</mn> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle Av=6v}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/428e73acd3e396c76ca972a09dbd8aba35ff4cc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.259ex; height:2.176ex;" alt="{\displaystyle Av=6v}">, that is <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}4&1\\6&3\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}=6\cdot {\begin{bmatrix}x\\y\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>4</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> </mtd> <mtd> <mn>3</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mn>6</mn> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mi>y</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}4&1\\6&3\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}=6\cdot {\begin{bmatrix}x\\y\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f21ef7f8c3ebb9b6428325a565af93be9a0cbb86" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.866ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}4&1\\6&3\end{bmatrix}}{\begin{bmatrix}x\\y\end{bmatrix}}=6\cdot {\begin{bmatrix}x\\y\end{bmatrix}}}"> This matrix equation is equivalent to two <a href="/wiki/Linear_equation" title="Linear equation">linear equations</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{aligned}4x+y&=6x\\6x+3y&=6y\end{aligned}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mn>4</mn> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>6</mn> <mi>x</mi> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> <mi>x</mi> <mo>+</mo> <mn>3</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>6</mn> <mi>y</mi> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}4x+y&=6x\\6x+3y&=6y\end{aligned}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f508a2712786db29fd7353690561b789bcba325c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.736ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{aligned}4x+y&=6x\\6x+3y&=6y\end{aligned}}\right.}">      that is      <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left\{{\begin{aligned}-2x+y&=0\\6x-3y&=0\end{aligned}}\right.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mo>−</mo> <mn>2</mn> <mi>x</mi> <mo>+</mo> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>6</mn> <mi>x</mi> <mo>−</mo> <mn>3</mn> <mi>y</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>0</mn> </mtd> </mtr> </mtable> </mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left\{{\begin{aligned}-2x+y&=0\\6x-3y&=0\end{aligned}}\right.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca289c65d959ddd158adad540bf0fefc08739c31" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.052ex; height:6.176ex;" alt="{\displaystyle \left\{{\begin{aligned}-2x+y&=0\\6x-3y&=0\end{aligned}}\right.}"> Both equations reduce to the single linear equation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=2x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> <mo>=</mo> <mn>2</mn> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y=2x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/582d8299e827a8ee042bff79fd37ead41199f7ef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.746ex; height:2.509ex;" alt="{\displaystyle y=2x}">. Therefore, any vector of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}a&2a\end{bmatrix}}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>a</mi> </mtd> <mtd> <mn>2</mn> <mi>a</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}a&2a\end{bmatrix}}^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/000f900a65bb62d01d6321e6b4181f1d48a083c5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.341ex; height:3.343ex;" alt="{\displaystyle {\begin{bmatrix}a&2a\end{bmatrix}}^{\textsf {T}}}">, for any nonzero real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}">, is an eigenvector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> with eigenvalue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =6}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mn>6</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =6}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/608b6a0bbc1583aea94c6ade00572a214789b6bd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =6}">. The matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}"> above has another eigenvalue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda =1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda =1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/543b4490416437b7c80ea473bbcac0e4ab7a7f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.616ex; height:2.176ex;" alt="{\displaystyle \lambda =1}">. A similar calculation shows that the corresponding eigenvectors are the nonzero solutions of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 3x+y=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>3</mn> <mi>x</mi> <mo>+</mo> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 3x+y=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c25f17e8f066fc07908117e5557786a6bb31820b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.749ex; height:2.509ex;" alt="{\displaystyle 3x+y=0}">, that is, any vector of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}b&-3b\end{bmatrix}}^{\textsf {T}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>b</mi> </mtd> <mtd> <mo>−</mo> <mn>3</mn> <mi>b</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mtext mathvariant="sans-serif">T</mtext> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}b&-3b\end{bmatrix}}^{\textsf {T}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7100afccdfb71f19da32467820f5d7d6e755ee2f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.685ex; height:3.343ex;" alt="{\displaystyle {\begin{bmatrix}b&-3b\end{bmatrix}}^{\textsf {T}}}">, for any nonzero real number <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>b</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle b}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.998ex; height:2.176ex;" alt="{\displaystyle b}">. <div class="mw-heading mw-heading3"><h3 id="Simple_iterative_methods">Simple iterative methods</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=32" title="Edit section: Simple iterative methods">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Power_iteration" title="Power iteration">Power iteration</a></div> The converse approach, of first seeking the eigenvectors and then determining each eigenvalue from its eigenvector, turns out to be far more tractable for computers. The easiest algorithm here consists of picking an arbitrary starting vector and then repeatedly multiplying it with the matrix (optionally normalizing the vector to keep its elements of reasonable size); this makes the vector converge towards an eigenvector. <a href="/wiki/Inverse_iteration" title="Inverse iteration">A variation</a> is to instead multiply the vector by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (A-\mu I)^{-1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>A</mi> <mo>−</mo> <mi>μ</mi> <mi>I</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (A-\mu I)^{-1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/030ec9b1fcaa93b58258707a8dab563cb6b693c4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.299ex; height:3.176ex;" alt="{\displaystyle (A-\mu I)^{-1}}">; this causes it to converge to an eigenvector of the eigenvalue closest to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu \in \mathbb {C} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>∈</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">C</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu \in \mathbb {C} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/677f38787c8576b04996f4edbf35e2c32af48706" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.92ex; height:2.676ex;" alt="{\displaystyle \mu \in \mathbb {C} }">. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }"> is (a good approximation of) an eigenvector of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, then the corresponding eigenvalue can be computed as <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ={\frac {\mathbf {v} ^{*}A\mathbf {v} }{\mathbf {v} ^{*}\mathbf {v} }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> <mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ={\frac {\mathbf {v} ^{*}A\mathbf {v} }{\mathbf {v} ^{*}\mathbf {v} }}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/42c3f719ad057237c1c44d571ba1599f91791865" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:10.909ex; height:5.343ex;" alt="{\displaystyle \lambda ={\frac {\mathbf {v} ^{*}A\mathbf {v} }{\mathbf {v} ^{*}\mathbf {v} }}}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} ^{*}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>∗</mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} ^{*}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/654d4509f8e6a5b8ee282fb3e4e981de66936466" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.465ex; height:2.343ex;" alt="{\displaystyle \mathbf {v} ^{*}}"> denotes the <a href="/wiki/Conjugate_transpose" title="Conjugate transpose">conjugate transpose</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.411ex; height:1.676ex;" alt="{\displaystyle \mathbf {v} }">. <div class="mw-heading mw-heading3"><h3 id="Modern_methods">Modern methods</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=33" title="Edit section: Modern methods">edit</a>]</div> Efficient, accurate methods to compute eigenvalues and eigenvectors of arbitrary matrices were not known until the <a href="/wiki/QR_algorithm" title="QR algorithm">QR algorithm</a> was designed in 1961.<a href="#cite_note-FOOTNOTETrefethenBau1997-47">[43]</a> Combining the <a href="/wiki/Householder_transformation" title="Householder transformation">Householder transformation</a> with the LU decomposition results in an algorithm with better convergence than the QR algorithm.[<a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed">citation needed</a>] For large <a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a> <a href="/wiki/Sparse_matrix" title="Sparse matrix">sparse matrices</a>, the <a href="/wiki/Lanczos_algorithm" title="Lanczos algorithm">Lanczos algorithm</a> is one example of an efficient <a href="/wiki/Iterative_method" title="Iterative method">iterative method</a> to compute eigenvalues and eigenvectors, among several other possibilities.<a href="#cite_note-FOOTNOTETrefethenBau1997-47">[43]</a> Most numeric methods that compute the eigenvalues of a matrix also determine a set of corresponding eigenvectors as a by-product of the computation, although sometimes implementors choose to discard the eigenvector information as soon as it is no longer needed. <div class="mw-heading mw-heading2"><h2 id="Applications">Applications</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=34" title="Edit section: Applications">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Geometric_transformations">Geometric transformations</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=35" title="Edit section: Geometric transformations">edit</a>]</div> Eigenvectors and eigenvalues can be useful for understanding linear transformations of geometric shapes. The following table presents some example transformations in the plane along with their 2×2 matrices, eigenvalues, and eigenvectors. <table class="wikitable" style="text-align:center; margin:1em auto 1em auto;"> <caption>Eigenvalues of geometric transformations </caption> <tbody><tr> <th> </th> <th scope="col"><a href="/wiki/Scaling_(geometry)" title="Scaling (geometry)">Scaling</a> </th> <th scope="col">Unequal scaling </th> <th scope="col"><a href="/wiki/Rotation_(geometry)" class="mw-redirect" title="Rotation (geometry)">Rotation</a> </th> <th scope="col"><a href="/wiki/Shear_mapping" title="Shear mapping">Horizontal shear</a> </th> <th scope="col"><a href="/wiki/Hyperbolic_rotation" class="mw-redirect" title="Hyperbolic rotation">Hyperbolic rotation</a> </th></tr> <tr> <th scope="row">Illustration </th> <td><a href="/wiki/File:Homothety_in_two_dim.svg" class="mw-file-description"><img alt="Equal scaling (homothety)" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Homothety_in_two_dim.svg/100px-Homothety_in_two_dim.svg.png" decoding="async" width="100" height="100" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Homothety_in_two_dim.svg/150px-Homothety_in_two_dim.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2b/Homothety_in_two_dim.svg/200px-Homothety_in_two_dim.svg.png 2x" data-file-width="1100" data-file-height="1100" /></a> </td> <td><a href="/wiki/File:Unequal_scaling.svg" class="mw-file-description"><img alt="Vertical shrink and horizontal stretch of a unit square." src="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Unequal_scaling.svg/100px-Unequal_scaling.svg.png" decoding="async" width="100" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/be/Unequal_scaling.svg/150px-Unequal_scaling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/be/Unequal_scaling.svg/200px-Unequal_scaling.svg.png 2x" data-file-width="800" data-file-height="600" /></a> </td> <td><a href="/wiki/File:Rotation.png" class="mw-file-description"><img alt="Rotation by 50 degrees" src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Rotation.png/100px-Rotation.png" decoding="async" width="100" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Rotation.png/150px-Rotation.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c9/Rotation.png/200px-Rotation.png 2x" data-file-width="303" data-file-height="299" /></a> </td> <td><figure class="mw-halign-center" typeof="mw:File"><a href="/wiki/File:Shear.svg" class="mw-file-description"><img alt="Horizontal shear mapping" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Shear.svg/100px-Shear.svg.png" decoding="async" width="100" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Shear.svg/150px-Shear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2a/Shear.svg/200px-Shear.svg.png 2x" data-file-width="800" data-file-height="600" /></a><figcaption></figcaption></figure> </td> <td><a href="/wiki/File:Squeeze_r%3D1.5.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Squeeze_r%3D1.5.svg/100px-Squeeze_r%3D1.5.svg.png" decoding="async" width="100" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Squeeze_r%3D1.5.svg/150px-Squeeze_r%3D1.5.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/67/Squeeze_r%3D1.5.svg/200px-Squeeze_r%3D1.5.svg.png 2x" data-file-width="820" data-file-height="550" /></a> </td></tr> <tr style="vertical-align:top"> <th scope="row">Matrix </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}k&0\\0&k\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>k</mi> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mi>k</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}k&0\\0&k\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e53e684e250ff477f3d46652383cae556aaf8eb8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.951ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}k&0\\0&k\end{bmatrix}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}k_{1}&0\\0&k_{2}\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}k_{1}&0\\0&k_{2}\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7bb5fc953bd28772a8946d5010c987f604198456" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.06ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}k_{1}&0\\0&k_{2}\end{bmatrix}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mo>−</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> <mtd> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/adf320fe078d6146945840b2ca5e6ae3ad7592b1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:16.646ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\cos \theta &-\sin \theta \\\sin \theta &\cos \theta \end{bmatrix}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}1&k\\0&1\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mi>k</mi> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}1&k\\0&1\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/91500c98e17498645a476b428553f65bc739829b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:7.903ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}1&k\\0&1\end{bmatrix}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{bmatrix}\cosh \varphi &\sinh \varphi \\\sinh \varphi &\cosh \varphi \end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <mi>sinh</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> <mtd> <mi>cosh</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{bmatrix}\cosh \varphi &\sinh \varphi \\\sinh \varphi &\cosh \varphi \end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6eafa5ef44c67ce595c7d848625e50ce3f99a9a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:18.151ex; height:6.176ex;" alt="{\displaystyle {\begin{bmatrix}\cosh \varphi &\sinh \varphi \\\sinh \varphi &\cosh \varphi \end{bmatrix}}}"> </td></tr> <tr> <th scope="row">Characteristic polynomial </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (\lambda -k)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <mi>k</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (\lambda -k)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0fdb919512cf0ff0f4009cf5c2e5628d036154cb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.851ex; height:3.176ex;" alt="{\displaystyle \ (\lambda -k)^{2}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (\lambda -k_{1})(\lambda -k_{2})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo stretchy="false">)</mo> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (\lambda -k_{1})(\lambda -k_{2})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/83039dceee60853050759e4ad606ca8a90068292" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.541ex; height:2.843ex;" alt="{\displaystyle (\lambda -k_{1})(\lambda -k_{2})}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{2}-2\cos(\theta )\lambda +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>cos</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>θ</mi> <mo stretchy="false">)</mo> <mi>λ</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{2}-2\cos(\theta )\lambda +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/199291ab40bf31b27e56a140cda9000fb666cea8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.168ex; height:3.176ex;" alt="{\displaystyle \lambda ^{2}-2\cos(\theta )\lambda +1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \ (\lambda -1)^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mtext> </mtext> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>−</mo> <mn>1</mn> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \ (\lambda -1)^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c87e8c25e09c5cdd0b21a172effb3f27efce3db9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.802ex; height:3.176ex;" alt="{\displaystyle \ (\lambda -1)^{2}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda ^{2}-2\cosh(\varphi )\lambda +1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>−</mo> <mn>2</mn> <mi>cosh</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>φ</mi> <mo stretchy="false">)</mo> <mi>λ</mi> <mo>+</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda ^{2}-2\cosh(\varphi )\lambda +1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6070a0113949b36fe2aabe1135511b74df871230" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.891ex; height:3.176ex;" alt="{\displaystyle \lambda ^{2}-2\cosh(\varphi )\lambda +1}"> </td></tr> <tr> <th scope="row">Eigenvalues, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/72fde940918edf84caf3d406cc7d31949166820f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle \lambda _{i}}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}=\lambda _{2}=k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}=\lambda _{2}=k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/326cbde7a4ba5352e8b638532df84753d1ec4b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.227ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}=\lambda _{2}=k}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\lambda _{1}&=k_{1}\\\lambda _{2}&=k_{2}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msub> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\lambda _{1}&=k_{1}\\\lambda _{2}&=k_{2}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ab13bedb0eb8f142ab85e007ed1a61fb65e348f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:8.525ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\lambda _{1}&=k_{1}\\\lambda _{2}&=k_{2}\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\lambda _{1}&=e^{i\theta }\\&=\cos \theta +i\sin \theta \\\lambda _{2}&=e^{-i\theta }\\&=\cos \theta -i\sin \theta \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>i</mi> <mi>θ</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>−</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\lambda _{1}&=e^{i\theta }\\&=\cos \theta +i\sin \theta \\\lambda _{2}&=e^{-i\theta }\\&=\cos \theta -i\sin \theta \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/711d76709080dbd1aa297b081a7de5140b654685" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:19.211ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\lambda _{1}&=e^{i\theta }\\&=\cos \theta +i\sin \theta \\\lambda _{2}&=e^{-i\theta }\\&=\cos \theta -i\sin \theta \end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda _{1}=\lambda _{2}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{1}=\lambda _{2}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8110a57bf3475c2bfba8d4af326160500b7bef6f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.178ex; height:2.509ex;" alt="{\displaystyle \lambda _{1}=\lambda _{2}=1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\lambda _{1}&=e^{\varphi }\\&=\cosh \varphi +\sinh \varphi \\\lambda _{2}&=e^{-\varphi }\\&=\cosh \varphi -\sinh \varphi \end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>φ</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>φ</mi> <mo>+</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> <mtr> <mtd> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <msup> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mi>φ</mi> </mrow> </msup> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>=</mo> <mi>cosh</mi> <mo>⁡</mo> <mi>φ</mi> <mo>−</mo> <mi>sinh</mi> <mo>⁡</mo> <mi>φ</mi> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\lambda _{1}&=e^{\varphi }\\&=\cosh \varphi +\sinh \varphi \\\lambda _{2}&=e^{-\varphi }\\&=\cosh \varphi -\sinh \varphi \end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9f070fa97214fe3800f7199ccb1e729e983bd5af" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:21.466ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\lambda _{1}&=e^{\varphi }\\&=\cosh \varphi +\sinh \varphi \\\lambda _{2}&=e^{-\varphi }\\&=\cosh \varphi -\sinh \varphi \end{aligned}}}"> </td></tr> <tr> <th scope="row">Algebraic <abbr title="multiplicity">mult.</abbr>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{i}=\mu (\lambda _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>μ</mi> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{i}=\mu (\lambda _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44a86bfbea85072f96882d88c90d4a1d8d42a7e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.666ex; height:2.843ex;" alt="{\displaystyle \mu _{i}=\mu (\lambda _{i})}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{1}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1}=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b804debefab063745106268754e2ad8f83a033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.717ex; height:2.676ex;" alt="{\displaystyle \mu _{1}=2}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0cc65133756ddac4dbf8fe2895482cac4301739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.468ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0cc65133756ddac4dbf8fe2895482cac4301739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.468ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{1}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{1}=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/37b804debefab063745106268754e2ad8f83a033" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.717ex; height:2.676ex;" alt="{\displaystyle \mu _{1}=2}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a0cc65133756ddac4dbf8fe2895482cac4301739" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.468ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\mu _{1}&=1\\\mu _{2}&=1\end{aligned}}}"> </td></tr> <tr> <th scope="row">Geometric <abbr title="multiplicity">mult.</abbr>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{i}=\gamma (\lambda _{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>γ</mi> <mo stretchy="false">(</mo> <msub> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{i}=\gamma (\lambda _{i})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ce3237dd9f6f5daba876c3e39baefcfbcca24504" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.329ex; height:2.843ex;" alt="{\displaystyle \gamma _{i}=\gamma (\lambda _{i})}"> </th> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}=2}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}=2}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e807587733866aeb774d767add51a536fe647f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.519ex; height:2.676ex;" alt="{\displaystyle \gamma _{1}=2}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50582dd278b30a4b8f7bbddc65fe886d5bc470f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.271ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50582dd278b30a4b8f7bbddc65fe886d5bc470f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.271ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma _{1}=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \gamma _{1}=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79b71e25e13c46dad7b2f6e07c3b6cf6e7a739d9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.519ex; height:2.676ex;" alt="{\displaystyle \gamma _{1}=1}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <msub> <mi>γ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/50582dd278b30a4b8f7bbddc65fe886d5bc470f9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:7.271ex; height:5.843ex;" alt="{\displaystyle {\begin{aligned}\gamma _{1}&=1\\\gamma _{2}&=1\end{aligned}}}"> </td></tr> <tr> <th scope="row">Eigenvectors </th> <td>All nonzero vectors </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\0\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}0\\1\end{bmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\0\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}0\\1\end{bmatrix}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7a1069ff8e7733cfde92189c7fff9b12ebd734e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:10.758ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\0\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}0\\1\end{bmatrix}}\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\-i\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\+i\end{bmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mi>i</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mi>i</mi> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\-i\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\+i\end{bmatrix}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebfcdafb92833dd29fcabb84de5349c305bf4626" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:12.207ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\-i\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\+i\end{bmatrix}}\end{aligned}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1}={\begin{bmatrix}1\\0\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} _{1}={\begin{bmatrix}1\\0\end{bmatrix}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/de1f984884c937803d30931a83c5310fefb97b5d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:10.007ex; height:6.176ex;" alt="{\displaystyle \mathbf {u} _{1}={\begin{bmatrix}1\\0\end{bmatrix}}}"> </td> <td><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\1\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\-1\end{bmatrix}}\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\1\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\-1\end{bmatrix}}\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea205646568b0a2e9d9c95c2d2920461a68992f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:12.567ex; height:12.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {u} _{1}&={\begin{bmatrix}1\\1\end{bmatrix}}\\\mathbf {u} _{2}&={\begin{bmatrix}1\\-1\end{bmatrix}}\end{aligned}}}"> </td></tr></tbody></table> The characteristic equation for a rotation is a <a href="/wiki/Quadratic_equation" title="Quadratic equation">quadratic equation</a> with <a href="/wiki/Discriminant" title="Discriminant">discriminant</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D=-4(\sin \theta )^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>=</mo> <mo>−</mo> <mn>4</mn> <mo stretchy="false">(</mo> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D=-4(\sin \theta )^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb5b1431621bd8cbae4165377fe93f7c4238840" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.19ex; height:3.176ex;" alt="{\displaystyle D=-4(\sin \theta )^{2}}">, which is a negative number whenever θ is not an integer multiple of 180°. Therefore, except for these special cases, the two eigenvalues are complex numbers, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \cos \theta \pm i\sin \theta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>±</mo> <mi>i</mi> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \cos \theta \pm i\sin \theta }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0f115a58c3b41db95de9a7e050e33a771ba9126e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.952ex; height:2.176ex;" alt="{\displaystyle \cos \theta \pm i\sin \theta }">; and all eigenvectors have non-real entries. Indeed, except for those special cases, a rotation changes the direction of every nonzero vector in the plane. A linear transformation that takes a square to a rectangle of the same area (a <a href="/wiki/Squeeze_mapping" title="Squeeze mapping">squeeze mapping</a>) has reciprocal eigenvalues. <div class="mw-heading mw-heading3"><h3 id="Principal_component_analysis">Principal component analysis</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=36" title="Edit section: Principal component analysis">edit</a>]</div> <figure class="mw-default-size mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:GaussianScatterPCA.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/GaussianScatterPCA.png/220px-GaussianScatterPCA.png" decoding="async" width="220" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/GaussianScatterPCA.png/330px-GaussianScatterPCA.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/GaussianScatterPCA.png/440px-GaussianScatterPCA.png 2x" data-file-width="1152" data-file-height="1081" /></a><figcaption>PCA of the <a href="/wiki/Multivariate_Gaussian_distribution" class="mw-redirect" title="Multivariate Gaussian distribution">multivariate Gaussian distribution</a> centered at <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (1,3)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>1</mn> <mo>,</mo> <mn>3</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (1,3)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c80e593e3953231c56d0887f5b247bbe517461f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.168ex; height:2.843ex;" alt="{\displaystyle (1,3)}"> with a standard deviation of 3 in roughly the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (0.878,0.478)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <mn>0.878</mn> <mo>,</mo> <mn>0.478</mn> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (0.878,0.478)}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/669ce2eb0449e640131848951e86653f1e13153c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.437ex; height:2.843ex;" alt="{\displaystyle (0.878,0.478)}"> direction and of 1 in the orthogonal direction. The vectors shown are unit eigenvectors of the (symmetric, positive-semidefinite) <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> scaled by the square root of the corresponding eigenvalue. Just as in the one-dimensional case, the square root is taken because the <a href="/wiki/Standard_deviation" title="Standard deviation">standard deviation</a> is more readily visualized than the <a href="/wiki/Variance" title="Variance">variance</a>.</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Principal_component_analysis" title="Principal component analysis">Principal component analysis</a></div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Positive_semidefinite_matrix" class="mw-redirect" title="Positive semidefinite matrix">Positive semidefinite matrix</a> and <a href="/wiki/Factor_analysis" title="Factor analysis">Factor analysis</a></div> The <a href="/wiki/Eigendecomposition_of_a_matrix#Real_symmetric_matrices" title="Eigendecomposition of a matrix">eigendecomposition</a> of a <a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> <a href="/wiki/Positive_semidefinite_matrix" class="mw-redirect" title="Positive semidefinite matrix">positive semidefinite</a> (PSD) <a href="/wiki/Positive_semidefinite_matrix" class="mw-redirect" title="Positive semidefinite matrix">matrix</a> yields an <a href="/wiki/Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a> of eigenvectors, each of which has a nonnegative eigenvalue. The orthogonal decomposition of a PSD matrix is used in <a href="/wiki/Multivariate_statistics" title="Multivariate statistics">multivariate analysis</a>, where the <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample</a> <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrices</a> are PSD. This orthogonal decomposition is called <a href="/wiki/Principal_component_analysis" title="Principal component analysis">principal component analysis</a> (PCA) in statistics. PCA studies <a href="/wiki/Linear_relation" title="Linear relation">linear relations</a> among variables. PCA is performed on the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> or the <a href="/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">correlation matrix</a> (in which each variable is scaled to have its <a href="/wiki/Sample_variance" class="mw-redirect" title="Sample variance">sample variance</a> equal to one). For the covariance or correlation matrix, the eigenvectors correspond to <a href="/wiki/Principal_component_analysis" title="Principal component analysis">principal components</a> and the eigenvalues to the <a href="/wiki/Explained_variance" class="mw-redirect" title="Explained variance">variance explained</a> by the principal components. Principal component analysis of the correlation matrix provides an <a href="/wiki/Orthogonal_basis" title="Orthogonal basis">orthogonal basis</a> for the space of the observed data: In this basis, the largest eigenvalues correspond to the principal components that are associated with most of the covariability among a number of observed data. Principal component analysis is used as a means of <a href="/wiki/Dimensionality_reduction" title="Dimensionality reduction">dimensionality reduction</a> in the study of large <a href="/wiki/Data_set" title="Data set">data sets</a>, such as those encountered in <a href="/wiki/Bioinformatics" title="Bioinformatics">bioinformatics</a>. In <a href="/wiki/Q_methodology" title="Q methodology">Q methodology</a>, the eigenvalues of the correlation matrix determine the Q-methodologist's judgment of practical significance (which differs from the <a href="/wiki/Statistical_significance" title="Statistical significance">statistical significance</a> of <a href="/wiki/Hypothesis_testing" class="mw-redirect" title="Hypothesis testing">hypothesis testing</a>; cf. <a href="/wiki/Scree%27s_test" class="mw-redirect" title="Scree's test">criteria for determining the number of factors</a>). More generally, principal component analysis can be used as a method of <a href="/wiki/Factor_analysis" title="Factor analysis">factor analysis</a> in <a href="/wiki/Structural_equation_model" class="mw-redirect" title="Structural equation model">structural equation modeling</a>. <div class="mw-heading mw-heading3"><h3 id="Graphs">Graphs</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=37" title="Edit section: Graphs">edit</a>]</div> In <a href="/wiki/Spectral_graph_theory" title="Spectral graph theory">spectral graph theory</a>, an eigenvalue of a <a href="/wiki/Graph_theory" title="Graph theory">graph</a> is defined as an eigenvalue of the graph's <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}">, or (increasingly) of the graph's <a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian matrix</a> due to its <a href="/wiki/Discrete_Laplace_operator" title="Discrete Laplace operator">discrete Laplace operator</a>, which is either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D-A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> <mo>−</mo> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D-A}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fad09e91e6bab106f78d995ed2eaa9deab02f446" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.508ex; height:2.343ex;" alt="{\displaystyle D-A}"> (sometimes called the combinatorial Laplacian) or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I-D^{-1/2}AD^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>I</mi> <mo>−</mo> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> <mi>A</mi> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle I-D^{-1/2}AD^{-1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4a684abb5b2846c23ec874bc8ec9bab43916f0a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:17.557ex; height:3.009ex;" alt="{\displaystyle I-D^{-1/2}AD^{-1/2}}"> (sometimes called the normalized Laplacian), where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>D</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f34a0c600395e5d4345287e21fb26efd386990e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}"> is a diagonal matrix with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{ii}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D_{ii}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4b72393211b47c0bc4b88090051f02efd3d9113" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.291ex; height:2.509ex;" alt="{\displaystyle D_{ii}}"> equal to the degree of vertex <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v_{i}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7dffe5726650f6daac54829972a94f38eb8ec127" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i}}">, and in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D^{-1/2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>D</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>−</mo> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle D^{-1/2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/84f8480690f195081acfe87f67dd044c7965fdb0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.901ex; height:2.843ex;" alt="{\displaystyle D^{-1/2}}">, the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}">th diagonal entry is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle 1/{\sqrt {\deg(v_{i})}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mn>1</mn> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>deg</mi> <mo>⁡</mo> <mo stretchy="false">(</mo> <msub> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle 1/{\sqrt {\deg(v_{i})}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3cd68f11bd7f7cbe229f0ef39a9e37a0d7b67fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:11.873ex; height:3.343ex;" alt="{\textstyle 1/{\sqrt {\deg(v_{i})}}}">. The <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">th principal eigenvector of a graph is defined as either the eigenvector corresponding to the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">th largest or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}">th smallest eigenvalue of the Laplacian. The first principal eigenvector of the graph is also referred to merely as the principal eigenvector. The principal eigenvector is used to measure the <a href="/wiki/Eigenvector_centrality" title="Eigenvector centrality">centrality</a> of its vertices. An example is <a href="/wiki/Google" title="Google">Google</a>'s <a href="/wiki/PageRank" title="PageRank">PageRank</a> algorithm. The principal eigenvector of a modified <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> of the World Wide Web graph gives the page ranks as its components. This vector corresponds to the <a href="/wiki/Stationary_distribution" title="Stationary distribution">stationary distribution</a> of the <a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a> represented by the row-normalized adjacency matrix; however, the adjacency matrix must first be modified to ensure a stationary distribution exists. The second smallest eigenvector can be used to partition the graph into clusters, via <a href="/wiki/Spectral_clustering" title="Spectral clustering">spectral clustering</a>. Other methods are also available for clustering. <div class="mw-heading mw-heading3"><h3 id="Markov_chains">Markov chains</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=38" title="Edit section: Markov chains">edit</a>]</div> A <a href="/wiki/Markov_chain" title="Markov chain">Markov chain</a> is represented by a matrix whose entries are the <a href="/wiki/Transition_probabilities" class="mw-redirect" title="Transition probabilities">transition probabilities</a> between states of a system. In particular the entries are non-negative, and every row of the matrix sums to one, being the sum of probabilities of transitions from one state to some other state of the system. The <a href="/wiki/Perron%E2%80%93Frobenius_theorem" title="Perron–Frobenius theorem">Perron–Frobenius theorem</a> gives sufficient conditions for a Markov chain to have a unique dominant eigenvalue, which governs the convergence of the system to a steady state. <div class="mw-heading mw-heading3"><h3 id="Vibration_analysis">Vibration analysis</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=39" title="Edit section: Vibration analysis">edit</a>]</div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif/220px-Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif" decoding="async" width="220" height="206" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/82/Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif/330px-Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/82/Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif/440px-Mode_Shape_of_a_Tuning_Fork_at_Eigenfrequency_440.09_Hz.gif 2x" data-file-width="656" data-file-height="614" /></a><figcaption>Mode shape of a tuning fork at eigenfrequency 440.09 Hz</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Vibration" title="Vibration">Vibration</a></div> Eigenvalue problems occur naturally in the vibration analysis of mechanical structures with many <a href="/wiki/Degrees_of_freedom_(mechanics)" title="Degrees of freedom (mechanics)">degrees of freedom</a>. The eigenvalues are the <a href="/wiki/Natural_frequency" title="Natural frequency">natural frequencies</a> (or eigenfrequencies) of vibration, and the eigenvectors are the shapes of these vibrational modes. In particular, undamped vibration is governed by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\ddot {x}}+kx=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>k</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\ddot {x}}+kx=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2159d09c7db1e7239c071a6219a42de08fd59f3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.012ex; height:2.343ex;" alt="{\displaystyle m{\ddot {x}}+kx=0}"> or <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\ddot {x}}=-kx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>=</mo> <mo>−</mo> <mi>k</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\ddot {x}}=-kx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/93228b5a83409e6b2ed3800bfa207d9d7c61d63f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:10.818ex; height:2.343ex;" alt="{\displaystyle m{\ddot {x}}=-kx}"> That is, acceleration is proportional to position (i.e., we expect <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/87f9e315fd7e2ba406057a97300593c4802b53e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}"> to be sinusoidal in time). In <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> dimensions, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"> becomes a <a href="/wiki/Mass_matrix" title="Mass matrix">mass matrix</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> a <a href="/wiki/Stiffness_matrix" title="Stiffness matrix">stiffness matrix</a>. Admissible solutions are then a linear combination of solutions to the <a href="/wiki/Generalized_eigenvalue_problem" class="mw-redirect" title="Generalized eigenvalue problem">generalized eigenvalue problem</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kx=\omega ^{2}mx}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mi>x</mi> <mo>=</mo> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle kx=\omega ^{2}mx}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e74e3d27b717e9bfc94d5b4f3762d892bdc58d0f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.51ex; height:2.676ex;" alt="{\displaystyle kx=\omega ^{2}mx}"> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega ^{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc60ab391d9835017f0778767fb25a54402d20f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.5ex; height:2.676ex;" alt="{\displaystyle \omega ^{2}}"> is the eigenvalue and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/48eff443f9de7a985bb94ca3bde20813ea737be8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }"> is the (imaginary) <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>. The principal <a href="/wiki/Vibration_mode" class="mw-redirect" title="Vibration mode">vibration modes</a> are different from the principal compliance modes, which are the eigenvectors of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}"> alone. Furthermore, <a href="/wiki/Damped_vibration" class="mw-redirect" title="Damped vibration">damped vibration</a>, governed by <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>¨</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mover> <mi>x</mi> <mo>˙</mo> </mover> </mrow> </mrow> <mo>+</mo> <mi>k</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94204a6eb83dcdc6550e0f4cad8b127dab620f4f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:18.189ex; height:2.343ex;" alt="{\displaystyle m{\ddot {x}}+c{\dot {x}}+kx=0}"> leads to a so-called <a href="/wiki/Quadratic_eigenvalue_problem" title="Quadratic eigenvalue problem">quadratic eigenvalue problem</a>, <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\omega ^{2}m+\omega c+k\right)x=0.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>m</mi> <mo>+</mo> <mi>ω</mi> <mi>c</mi> <mo>+</mo> <mi>k</mi> </mrow> <mo>)</mo> </mrow> <mi>x</mi> <mo>=</mo> <mn>0.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\omega ^{2}m+\omega c+k\right)x=0.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4ead2ed546c17096d1e8537e3d24c56c65db0e7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:22.639ex; height:3.343ex;" alt="{\displaystyle \left(\omega ^{2}m+\omega c+k\right)x=0.}"> This can be reduced to a generalized eigenvalue problem by <a href="/wiki/Quadratic_eigenvalue_problem#Methods_of_Solution" title="Quadratic eigenvalue problem">algebraic manipulation</a> at the cost of solving a larger system. The orthogonality properties of the eigenvectors allows decoupling of the <a href="/wiki/Differential_equation" title="Differential equation">differential equations</a> so that the system can be represented as linear summation of the eigenvectors. The eigenvalue problem of complex structures is often solved using <a href="/wiki/Finite_element_analysis" class="mw-redirect" title="Finite element analysis">finite element analysis</a>, but neatly generalize the solution to scalar-valued vibration problems. <div class="mw-heading mw-heading3"><h3 id="Tensor_of_moment_of_inertia">Tensor of moment of inertia</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=40" title="Edit section: Tensor of moment of inertia">edit</a>]</div> In <a href="/wiki/Mechanics" title="Mechanics">mechanics</a>, the eigenvectors of the <a href="/wiki/Inertia_tensor" class="mw-redirect" title="Inertia tensor">moment of inertia tensor</a> define the <a href="/wiki/Principal_axis_(mechanics)" class="mw-redirect" title="Principal axis (mechanics)">principal axes</a> of a <a href="/wiki/Rigid_body" title="Rigid body">rigid body</a>. The <a href="/wiki/Tensor" title="Tensor">tensor</a> of moment of <a href="/wiki/Inertia" title="Inertia">inertia</a> is a key quantity required to determine the rotation of a rigid body around its <a href="/wiki/Center_of_mass" title="Center of mass">center of mass</a>. <div class="mw-heading mw-heading3"><h3 id="Stress_tensor">Stress tensor</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=41" title="Edit section: Stress tensor">edit</a>]</div> In <a href="/wiki/Solid_mechanics" title="Solid mechanics">solid mechanics</a>, the <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> tensor is symmetric and so can be decomposed into a <a href="/wiki/Diagonal" title="Diagonal">diagonal</a> tensor with the eigenvalues on the diagonal and eigenvectors as a basis. Because it is diagonal, in this orientation, the stress tensor has no <a href="/wiki/Shear_(mathematics)" class="mw-redirect" title="Shear (mathematics)">shear</a> components; the components it does have are the principal components. <div class="mw-heading mw-heading3"><h3 id="Schrödinger_equation">Schrödinger equation</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=42" title="Edit section: Schrödinger equation">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:HAtomOrbitals.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cf/HAtomOrbitals.png/271px-HAtomOrbitals.png" decoding="async" width="271" height="271" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/c/cf/HAtomOrbitals.png 1.5x" data-file-width="316" data-file-height="316" /></a><figcaption>The <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunctions</a> associated with the <a href="/wiki/Bound_state" title="Bound state">bound states</a> of an <a href="/wiki/Electron" title="Electron">electron</a> in a <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom</a> can be seen as the eigenvectors of the <a href="/wiki/Hydrogen_atom" title="Hydrogen atom">hydrogen atom Hamiltonian</a> as well as of the <a href="/wiki/Angular_momentum_operator" title="Angular momentum operator">angular momentum operator</a>. They are associated with eigenvalues interpreted as their energies (increasing downward: <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n=1,\,2,\,3,\,\ldots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>2</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mn>3</mn> <mo>,</mo> <mspace width="thinmathspace" /> <mo>…</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n=1,\,2,\,3,\,\ldots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d7aea7d9828557e2300f79018dcf2f879fa47ab6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.967ex; height:2.509ex;" alt="{\displaystyle n=1,\,2,\,3,\,\ldots }">) and <a href="/wiki/Angular_momentum" title="Angular momentum">angular momentum</a> (increasing across: s, p, d, ...). The illustration shows the square of the absolute value of the wavefunctions. Brighter areas correspond to higher <a href="/wiki/Probability_density_function" title="Probability density function">probability density</a> for a position <a href="/wiki/Measurement_in_quantum_mechanics" title="Measurement in quantum mechanics">measurement</a>. The center of each figure is the <a href="/wiki/Atomic_nucleus" title="Atomic nucleus">atomic nucleus</a>, a <a href="/wiki/Proton" title="Proton">proton</a>.</figcaption></figure> An example of an eigenvalue equation where the transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle T}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>T</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}"> is represented in terms of a differential operator is the time-independent <a href="/wiki/Schr%C3%B6dinger_equation" title="Schrödinger equation">Schrödinger equation</a> in <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\psi _{E}=E\psi _{E}\,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo>=</mo> <mi>E</mi> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mspace width="thinmathspace" /> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\psi _{E}=E\psi _{E}\,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2bee1b56b8c500f5d904ad07ed6c310652d20098" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.327ex; height:2.509ex;" alt="{\displaystyle H\psi _{E}=E\psi _{E}\,}"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}">, the <a href="/wiki/Hamiltonian_(quantum_mechanics)" title="Hamiltonian (quantum mechanics)">Hamiltonian</a>, is a second-order <a href="/wiki/Differential_operator" title="Differential operator">differential operator</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{E}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d591fcd4d776cad11073ce439ba393b0fa15b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.001ex; height:2.509ex;" alt="{\displaystyle \psi _{E}}">, the <a href="/wiki/Wavefunction" class="mw-redirect" title="Wavefunction">wavefunction</a>, is one of its eigenfunctions corresponding to the eigenvalue <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}">, interpreted as its <a href="/wiki/Energy" title="Energy">energy</a>. However, in the case where one is interested only in the <a href="/wiki/Bound_state" title="Bound state">bound state</a> solutions of the Schrödinger equation, one looks for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{E}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d591fcd4d776cad11073ce439ba393b0fa15b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.001ex; height:2.509ex;" alt="{\displaystyle \psi _{E}}"> within the space of <a href="/wiki/Square-integrable_function" title="Square-integrable function">square integrable</a> functions. Since this space is a <a href="/wiki/Hilbert_space" title="Hilbert space">Hilbert space</a> with a well-defined <a href="/wiki/Scalar_product" class="mw-redirect" title="Scalar product">scalar product</a>, one can introduce a <a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">basis set</a> in which <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \psi _{E}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \psi _{E}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4d591fcd4d776cad11073ce439ba393b0fa15b6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.001ex; height:2.509ex;" alt="{\displaystyle \psi _{E}}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> can be represented as a one-dimensional array (i.e., a vector) and a matrix respectively. This allows one to represent the Schrödinger equation in a matrix form. The <a href="/wiki/Bra%E2%80%93ket_notation" title="Bra–ket notation">bra–ket notation</a> is often used in this context. A vector, which represents a state of the system, in the Hilbert space of square integrable functions is represented by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi _{E}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi _{E}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e76cd2a3ae07928b0d956dc5d35cc2cfc1fb25b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.847ex; height:2.843ex;" alt="{\displaystyle |\Psi _{E}\rangle }">. In this notation, the Schrödinger equation is: <dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H|\Psi _{E}\rangle =E|\Psi _{E}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> <mo>=</mo> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H|\Psi _{E}\rangle =E|\Psi _{E}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/629f94a4338f4b85c597313e6f194135f7b6c751" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.633ex; height:2.843ex;" alt="{\displaystyle H|\Psi _{E}\rangle =E|\Psi _{E}\rangle }"></dd></dl> where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi _{E}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi _{E}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e76cd2a3ae07928b0d956dc5d35cc2cfc1fb25b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.847ex; height:2.843ex;" alt="{\displaystyle |\Psi _{E}\rangle }"> is an eigenstate of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>E</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}"> represents the eigenvalue. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> is an <a href="/wiki/Observable" title="Observable">observable</a> <a href="/wiki/Self-adjoint_operator" title="Self-adjoint operator">self-adjoint operator</a>, the infinite-dimensional analog of Hermitian matrices. As in the matrix case, in the equation above <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H|\Psi _{E}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H|\Psi _{E}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3da526400f9a3bf744ce3fb63fe6c3c5452f3841" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.911ex; height:2.843ex;" alt="{\displaystyle H|\Psi _{E}\rangle }"> is understood to be the vector obtained by application of the transformation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/75a9edddcca2f782014371f75dca39d7e13a9c1b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\Psi _{E}\rangle }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mi mathvariant="normal">Ψ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>E</mi> </mrow> </msub> <mo fence="false" stretchy="false">⟩</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle |\Psi _{E}\rangle }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3e76cd2a3ae07928b0d956dc5d35cc2cfc1fb25b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.847ex; height:2.843ex;" alt="{\displaystyle |\Psi _{E}\rangle }">. <div class="mw-heading mw-heading3"><h3 id="Wave_transport">Wave transport</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=43" title="Edit section: Wave transport">edit</a>]</div> <a href="/wiki/Light" title="Light">Light</a>, <a href="/wiki/Acoustic_wave" title="Acoustic wave">acoustic waves</a>, and <a href="/wiki/Microwave" title="Microwave">microwaves</a> are randomly <a href="/wiki/Scattering_theory" class="mw-redirect" title="Scattering theory">scattered</a> numerous times when traversing a static <a href="/w/index.php?title=Disordered_system&action=edit&redlink=1" class="new" title="Disordered system (page does not exist)">disordered system</a>. Even though multiple scattering repeatedly randomizes the waves, ultimately coherent wave transport through the system is a deterministic process which can be described by a field transmission matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {t} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {t} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/16ff63cc74a931900e79b3caacbae3fa8cc66845" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.039ex; height:2.009ex;" alt="{\displaystyle \mathbf {t} }">.<a href="#cite_note-FOOTNOTEVellekoopMosk20072309–2311-49">[44]</a><a href="#cite_note-FOOTNOTERotterGigan201715005-50">[45]</a> The eigenvectors of the transmission operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d261a382e89e7d3da784b96ba0c22cf45f87666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.041ex; height:2.676ex;" alt="{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }"> form a set of disorder-specific input wavefronts which enable waves to couple into the disordered system's eigenchannels: the independent pathways waves can travel through the system. The eigenvalues, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>τ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }">, of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>†</mo> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">t</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d261a382e89e7d3da784b96ba0c22cf45f87666" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.041ex; height:2.676ex;" alt="{\displaystyle \mathbf {t} ^{\dagger }\mathbf {t} }"> correspond to the intensity transmittance associated with each eigenchannel. One of the remarkable properties of the transmission operator of diffusive systems is their bimodal eigenvalue distribution with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{\max }=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">max</mo> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\max }=1}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f09aaa446beea3630cbfc1d5ed5e7fe5183d1d53" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.568ex; height:2.509ex;" alt="{\displaystyle \tau _{\max }=1}"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{\min }=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo movablelimits="true" form="prefix">min</mo> </mrow> </msub> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{\min }=0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/63a24a70d3df1842251609baa582b5dd466165db" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.25ex; height:2.509ex;" alt="{\displaystyle \tau _{\min }=0}">.<a href="#cite_note-FOOTNOTERotterGigan201715005-50">[45]</a> Furthermore, one of the striking properties of open eigenchannels, beyond the perfect transmittance, is the statistically robust spatial profile of the eigenchannels.<a href="#cite_note-FOOTNOTEBenderYamilovYilmazCao2020165901-51">[46]</a> <div class="mw-heading mw-heading3"><h3 id="Molecular_orbitals">Molecular orbitals</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=44" title="Edit section: Molecular orbitals">edit</a>]</div> In <a href="/wiki/Quantum_mechanics" title="Quantum mechanics">quantum mechanics</a>, and in particular in <a href="/wiki/Atomic_physics" title="Atomic physics">atomic</a> and <a href="/wiki/Molecular_physics" title="Molecular physics">molecular physics</a>, within the <a href="/wiki/Hartree%E2%80%93Fock" class="mw-redirect" title="Hartree–Fock">Hartree–Fock</a> theory, the <a href="/wiki/Atomic_orbital" title="Atomic orbital">atomic</a> and <a href="/wiki/Molecular_orbital" title="Molecular orbital">molecular orbitals</a> can be defined by the eigenvectors of the <a href="/wiki/Fock_operator" class="mw-redirect" title="Fock operator">Fock operator</a>. The corresponding eigenvalues are interpreted as <a href="/wiki/Ionization_potential" class="mw-redirect" title="Ionization potential">ionization potentials</a> via <a href="/wiki/Koopmans%27_theorem" title="Koopmans' theorem">Koopmans' theorem</a>. In this case, the term eigenvector is used in a somewhat more general meaning, since the Fock operator is explicitly dependent on the orbitals and their eigenvalues. Thus, if one wants to underline this aspect, one speaks of nonlinear eigenvalue problems. Such equations are usually solved by an <a href="/wiki/Iteration" title="Iteration">iteration</a> procedure, called in this case <a href="/wiki/Self-consistent_field" class="mw-redirect" title="Self-consistent field">self-consistent field</a> method. In <a href="/wiki/Quantum_chemistry" title="Quantum chemistry">quantum chemistry</a>, one often represents the Hartree–Fock equation in a non-<a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a> <a href="/wiki/Basis_set_(chemistry)" title="Basis set (chemistry)">basis set</a>. This particular representation is a <a href="/wiki/Generalized_eigenvalue_problem" class="mw-redirect" title="Generalized eigenvalue problem">generalized eigenvalue problem</a> called <a href="/wiki/Roothaan_equations" title="Roothaan equations">Roothaan equations</a>. <div class="mw-heading mw-heading3"><h3 id="Geology_and_glaciology">Geology and glaciology</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=45" title="Edit section: Geology and glaciology">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1251242444">.mw-parser-output .ambox{border:1px solid #a2a9b1;border-left:10px solid #36c;background-color:#fbfbfb;box-sizing:border-box}.mw-parser-output .ambox+link+.ambox,.mw-parser-output .ambox+link+style+.ambox,.mw-parser-output .ambox+link+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+style+.ambox,.mw-parser-output .ambox+.mw-empty-elt+link+link+.ambox{margin-top:-1px}html body.mediawiki .mw-parser-output .ambox.mbox-small-left{margin:4px 1em 4px 0;overflow:hidden;width:238px;border-collapse:collapse;font-size:88%;line-height:1.25em}.mw-parser-output .ambox-speedy{border-left:10px solid #b32424;background-color:#fee7e6}.mw-parser-output .ambox-delete{border-left:10px solid #b32424}.mw-parser-output .ambox-content{border-left:10px solid #f28500}.mw-parser-output .ambox-style{border-left:10px solid #fc3}.mw-parser-output .ambox-move{border-left:10px solid #9932cc}.mw-parser-output .ambox-protection{border-left:10px solid #a2a9b1}.mw-parser-output .ambox .mbox-text{border:none;padding:0.25em 0.5em;width:100%}.mw-parser-output .ambox .mbox-image{border:none;padding:2px 0 2px 0.5em;text-align:center}.mw-parser-output .ambox .mbox-imageright{border:none;padding:2px 0.5em 2px 0;text-align:center}.mw-parser-output .ambox .mbox-empty-cell{border:none;padding:0;width:1px}.mw-parser-output .ambox .mbox-image-div{width:52px}@media(min-width:720px){.mw-parser-output .ambox{margin:0 10%}}@media print{body.ns-0 .mw-parser-output .ambox{display:none!important}}</style><table class="box-Technical plainlinks metadata ambox ambox-style ambox-technical" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This section may be too technical for most readers to understand. Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit">help improve it</a> to <a href="/wiki/Wikipedia:Make_technical_articles_understandable" title="Wikipedia:Make technical articles understandable">make it understandable to non-experts</a>, without removing the technical details. (December 2023) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> In <a href="/wiki/Geology" title="Geology">geology</a>, especially in the study of <a href="/wiki/Glacial_till" class="mw-redirect" title="Glacial till">glacial till</a>, eigenvectors and eigenvalues are used as a method by which a mass of information of a clast fabric's constituents' orientation and dip can be summarized in a 3-D space by six numbers. In the field, a geologist may collect such data for hundreds or thousands of <a href="/wiki/Clasts" class="mw-redirect" title="Clasts">clasts</a> in a soil sample, which can only be compared graphically such as in a Tri-Plot (Sneed and Folk) diagram,<a href="#cite_note-FOOTNOTEGrahamMidgley20001473–1477-52">[47]</a><a href="#cite_note-FOOTNOTESneedFolk1958114–150-53">[48]</a> or as a Stereonet on a Wulff Net.<a href="#cite_note-FOOTNOTEKnox-RobinsonGardoll1998243-54">[49]</a> The output for the orientation tensor is in the three orthogonal (perpendicular) axes of space. The three eigenvectors are ordered <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\mathbf {v} _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\mathbf {v} _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a16fb872d837aeb0e6b7e62a458f6f0cc6c5f11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.464ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\mathbf {v} _{3}}"> by their eigenvalues <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}\geq E_{2}\geq E_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>≥</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}\geq E_{2}\geq E_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d8f97f04075af676d7b439ce24e6295806f4042" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.505ex; height:2.509ex;" alt="{\displaystyle E_{1}\geq E_{2}\geq E_{3}}">;<a href="#cite_note-55">[50]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282458bb19c231f94697405bddd93af04a34cabe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.465ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1}}"> then is the primary orientation/dip of clast, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/498720fbe6f897f2b86d2cf0f37498d682932aa0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.465ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{2}}"> is the secondary and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">v</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {v} _{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edbc3fb88d6a27517dc79da85446ab8147801be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.465ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{3}}"> is the tertiary, in terms of strength. The clast orientation is defined as the direction of the eigenvector, on a <a href="/wiki/Compass_rose" title="Compass rose">compass rose</a> of <a href="/wiki/Turn_(geometry)" class="mw-redirect" title="Turn (geometry)">360°</a>. Dip is measured as the eigenvalue, the modulus of the tensor: this is valued from 0° (no dip) to 90° (vertical). The relative values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ac42446bcd2cbb76ec8fe2895635d328da22e26" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{1}}">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{2}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e6ee346e54f38302f47b5cf3016d8718f2040c0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{2}}">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/395f570db7ae98f5a9a0d7b8436ae8c61ce5ebdc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.769ex; height:2.509ex;" alt="{\displaystyle E_{3}}"> are dictated by the nature of the sediment's fabric. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}=E_{2}=E_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}=E_{2}=E_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/44fe14484e78be8a197227a376449a236403e49d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.505ex; height:2.509ex;" alt="{\displaystyle E_{1}=E_{2}=E_{3}}">, the fabric is said to be isotropic. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}=E_{2}>E_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}=E_{2}>E_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b0d25ff62dc33b187f2b73e39bda3c972724aa2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.505ex; height:2.509ex;" alt="{\displaystyle E_{1}=E_{2}>E_{3}}">, the fabric is said to be planar. If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{1}>E_{2}>E_{3}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>></mo> <msub> <mi>E</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle E_{1}>E_{2}>E_{3}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/215bda2a0fb9f5ccbbb7ada013a2d3fb8d82150c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.505ex; height:2.509ex;" alt="{\displaystyle E_{1}>E_{2}>E_{3}}">, the fabric is said to be linear.<a href="#cite_note-FOOTNOTEBennEvans2004103–107-56">[51]</a> <div class="mw-heading mw-heading3"><h3 id="Basic_reproduction_number">Basic reproduction number</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=46" title="Edit section: Basic reproduction number">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Basic_reproduction_number" title="Basic reproduction number">Basic reproduction number</a></div> The basic reproduction number (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8916196f182fcbaaca54f931176a4a4f5769cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{0}}">) is a fundamental number in the study of how infectious diseases spread. If one infectious person is put into a population of completely susceptible people, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8916196f182fcbaaca54f931176a4a4f5769cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{0}}"> is the average number of people that one typical infectious person will infect. The generation time of an infection is the time, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{G}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{G}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a6fe2ac27c2455c992b9b30de7063223a9f6f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.364ex; height:2.343ex;" alt="{\displaystyle t_{G}}">, from one person becoming infected to the next person becoming infected. In a heterogeneous population, the next generation matrix defines how many people in the population will become infected after time <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t_{G}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>G</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t_{G}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/07a6fe2ac27c2455c992b9b30de7063223a9f6f2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.364ex; height:2.343ex;" alt="{\displaystyle t_{G}}"> has passed. The value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R_{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R_{0}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b8916196f182fcbaaca54f931176a4a4f5769cc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.818ex; height:2.509ex;" alt="{\displaystyle R_{0}}"> is then the largest eigenvalue of the next generation matrix.<a href="#cite_note-FOOTNOTEDiekmannHeesterbeekMetz1990365–382-57">[52]</a><a href="#cite_note-FOOTNOTEHeesterbeekDiekmann2000-58">[53]</a> <div class="mw-heading mw-heading3"><h3 id="Eigenfaces">Eigenfaces</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=47" title="Edit section: Eigenfaces">edit</a>]</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Eigenfaces.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Eigenfaces.png/200px-Eigenfaces.png" decoding="async" width="200" height="239" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/67/Eigenfaces.png/300px-Eigenfaces.png 1.5x, //upload.wikimedia.org/wikipedia/commons/6/67/Eigenfaces.png 2x" data-file-width="357" data-file-height="426" /></a><figcaption><a href="/wiki/Eigenface" title="Eigenface">Eigenfaces</a> as examples of eigenvectors</figcaption></figure> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1236090951"><div role="note" class="hatnote navigation-not-searchable">Main article: <a href="/wiki/Eigenface" title="Eigenface">Eigenface</a></div> In <a href="/wiki/Image_processing" class="mw-redirect" title="Image processing">image processing</a>, processed images of faces can be seen as vectors whose components are the <a href="/wiki/Brightness" title="Brightness">brightnesses</a> of each <a href="/wiki/Pixel" title="Pixel">pixel</a>.<a href="#cite_note-FOOTNOTEXirouhakisVotsisDelopoulus2004-59">[54]</a> The dimension of this vector space is the number of pixels. The eigenvectors of the <a href="/wiki/Covariance_matrix" title="Covariance matrix">covariance matrix</a> associated with a large set of normalized pictures of faces are called <a href="/wiki/Eigenface" title="Eigenface">eigenfaces</a>; this is an example of <a href="/wiki/Principal_component_analysis" title="Principal component analysis">principal component analysis</a>. They are very useful for expressing any face image as a <a href="/wiki/Linear_combination" title="Linear combination">linear combination</a> of some of them. In the <a href="/wiki/Facial_recognition_system" title="Facial recognition system">facial recognition</a> branch of <a href="/wiki/Biometrics" title="Biometrics">biometrics</a>, eigenfaces provide a means of applying <a href="/wiki/Data_compression" title="Data compression">data compression</a> to faces for <a href="/wiki/Recognition_of_human_individuals" class="mw-redirect" title="Recognition of human individuals">identification</a> purposes. Research related to eigen vision systems determining hand gestures has also been made. Similar to this concept, eigenvoices represent the general direction of variability in human pronunciations of a particular utterance, such as a word in a language. Based on a linear combination of such eigenvoices, a new voice pronunciation of the word can be constructed. These concepts have been found useful in automatic speech recognition systems for speaker adaptation. <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=48" title="Edit section: See also">edit</a>]</div> <ul><li><a href="/wiki/Antieigenvalue_theory" title="Antieigenvalue theory">Antieigenvalue theory</a></li> <li><a href="/wiki/Eigenoperator" title="Eigenoperator">Eigenoperator</a></li> <li><a href="/wiki/Eigenplane" title="Eigenplane">Eigenplane</a></li> <li><a href="/wiki/Eigenmoments" title="Eigenmoments">Eigenmoments</a></li> <li><a href="/wiki/Eigenvalue_algorithm" title="Eigenvalue algorithm">Eigenvalue algorithm</a></li> <li><a href="/wiki/Quantum_states" class="mw-redirect" title="Quantum states">Quantum states</a></li> <li><a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a></li> <li><a href="/wiki/List_of_numerical-analysis_software" title="List of numerical-analysis software">List of numerical-analysis software</a></li> <li><a href="/wiki/Nonlinear_eigenproblem" title="Nonlinear eigenproblem">Nonlinear eigenproblem</a></li> <li><a href="/wiki/Normal_eigenvalue" title="Normal eigenvalue">Normal eigenvalue</a></li> <li><a href="/wiki/Quadratic_eigenvalue_problem" title="Quadratic eigenvalue problem">Quadratic eigenvalue problem</a></li> <li><a href="/wiki/Singular_value" title="Singular value">Singular value</a></li> <li><a href="/wiki/Spectrum_of_a_matrix" title="Spectrum of a matrix">Spectrum of a matrix</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=49" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-lower-alpha"> <div class="mw-references-wrap"><ol class="references"> <li id="cite_note-10"><a href="#cite_ref-10">^</a> Note: <ul><li>In 1751, Leonhard Euler proved that any body has a principal axis of rotation: Leonhard Euler (presented: October 1751; published: 1760) <a rel="nofollow" class="external text" href="https://archive.org/stream/histoiredelacad07unkngoog#page/n196/mode/2up">"Du mouvement d'un corps solide quelconque lorsqu'il tourne autour d'un axe mobile"</a> (On the movement of any solid body while it rotates around a moving axis), Histoire de l'Académie royale des sciences et des belles lettres de Berlin, pp. 176–227. <a rel="nofollow" class="external text" href="https://archive.org/stream/histoiredelacad07unkngoog#page/n232/mode/2up">On p. 212</a>, Euler proves that any body contains a principal axis of rotation: "Théorem. 44. De quelque figure que soit le corps, on y peut toujours assigner un tel axe, qui passe par son centre de gravité, autour duquel le corps peut tourner librement & d'un mouvement uniforme." (Theorem. 44. Whatever be the shape of the body, one can always assign to it such an axis, which passes through its center of gravity, around which it can rotate freely and with a uniform motion.)</li> <li>In 1755, <a href="/wiki/Johann_Andreas_Segner" title="Johann Andreas Segner">Johann Andreas Segner</a> proved that any body has three principal axes of rotation: Johann Andreas Segner, Specimen theoriae turbinum [Essay on the theory of tops (i.e., rotating bodies)] ( Halle ("Halae"), (Germany): Gebauer, 1755). (<a rel="nofollow" class="external free" href="https://books.google.com/books?id=29">https://books.google.com/books?id=29</a> p. xxviiii [29]), Segner derives a third-degree equation in t, which proves that a body has three principal axes of rotation. He then states (on the same page): "Non autem repugnat tres esse eiusmodi positiones plani HM, quia in aequatione cubica radices tres esse possunt, et tres tangentis t valores." (However, it is not inconsistent [that there] be three such positions of the plane HM, because in cubic equations, [there] can be three roots, and three values of the tangent t.)</li> <li>The relevant passage of Segner's work was discussed briefly by <a href="/wiki/Arthur_Cayley" title="Arthur Cayley">Arthur Cayley</a>. See: A. Cayley (1862) "Report on the progress of the solution of certain special problems of dynamics," Report of the Thirty-second meeting of the British Association for the Advancement of Science; held at Cambridge in October 1862, 32: 184–252; see especially <a rel="nofollow" class="external text" href="https://books.google.com/books?id=S_RJAAAAcAAJ&pg=PA225">pp. 225–226.</a></li></ul> </li> <li id="cite_note-13"><a href="#cite_ref-13">^</a> <a href="#CITEREFKline1972">Kline 1972</a>, pp. 807–808 Augustin Cauchy (1839) "Mémoire sur l'intégration des équations linéaires" (Memoir on the integration of linear equations), Comptes rendus, 8: 827–830, 845–865, 889–907, 931–937. <a rel="nofollow" class="external text" href="https://gallica.bnf.fr/ark:/12148/bpt6k2967c/f833.item.r=.zoom">From p. 827:</a> "On sait d'ailleurs qu'en suivant la méthode de Lagrange, on obtient pour valeur générale de la variable prinicipale une fonction dans laquelle entrent avec la variable principale les racines d'une certaine équation que j'appellerai l'équation caractéristique, le degré de cette équation étant précisément l'order de l'équation différentielle qu'il s'agit d'intégrer." (One knows, moreover, that by following Lagrange's method, one obtains for the general value of the principal variable a function in which there appear, together with the principal variable, the roots of a certain equation that I will call the "characteristic equation", the degree of this equation being precisely the order of the differential equation that must be integrated.) </li> <li id="cite_note-19"><a href="#cite_ref-19">^</a> See: <ul><li>David Hilbert (1904) <a rel="nofollow" class="external text" href="https://digizeitschriften.de/dms/img/?PPN=PPN252457811_1904&DMDID=dmdlog11&LOGID=log11&PHYSID=phys57#navi">"Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (Erste Mitteilung)"</a> (Fundamentals of a general theory of linear integral equations. (First report)), Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse (News of the Philosophical Society at Göttingen, mathematical-physical section), pp. 49–91. <a rel="nofollow" class="external text" href="https://digizeitschriften.de/dms/img/?PPN=PPN252457811_1904&DMDID=dmdlog11&LOGID=log11&PHYSID=phys57#navi">From p. 51:</a> "Insbesondere in dieser ersten Mitteilung gelange ich zu Formeln, die die Entwickelung einer willkürlichen Funktion nach gewissen ausgezeichneten Funktionen, die ich 'Eigenfunktionen' nenne, liefern: ..." (In particular, in this first report I arrive at formulas that provide the [series] development of an arbitrary function in terms of some distinctive functions, which I call eigenfunctions: ... ) Later on the same page: "Dieser Erfolg ist wesentlich durch den Umstand bedingt, daß ich nicht, wie es bisher geschah, in erster Linie auf den Beweis für die Existenz der Eigenwerte ausgehe, ... " (This success is mainly attributable to the fact that I do not, as it has happened until now, first of all aim at a proof of the existence of eigenvalues...)</li> <li>For the origin and evolution of the terms eigenvalue, characteristic value, etc., see: <a rel="nofollow" class="external text" href="https://jeff560.tripod.com/e.html">Earliest Known Uses of Some of the Words of Mathematics (E)</a></li></ul> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> For a proof of this lemma, see <a href="#CITEREFRoman2008">Roman 2008</a>, Theorem 8.2 on p. 186; <a href="#CITEREFShilov1977">Shilov 1977</a>, p. 109; <a href="#CITEREFHefferon2001">Hefferon 2001</a>, p. 364; <a href="#CITEREFBeezer2006">Beezer 2006</a>, Theorem EDELI on p. 469; and <a href="https://en.wikibooks.org/wiki/Famous_Theorems_of_Mathematics/Algebra/Linear_Transformations#Lemma_for_linear_independence_of_eigenvectors" class="extiw" title="b:Famous Theorems of Mathematics/Algebra/Linear Transformations">Lemma for linear independence of eigenvectors</a> </li> <li id="cite_note-48"><a href="#cite_ref-48">^</a> By doing <a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a> over <a href="/wiki/Formal_power_series" title="Formal power series">formal power series</a> truncated to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}"> terms it is possible to get away with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(n^{4})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>O</mi> <mo stretchy="false">(</mo> <msup> <mi>n</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle O(n^{4})}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ae98b94e039c8eafabf6fe6a0128d232ed7d21f6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.032ex; height:3.176ex;" alt="{\displaystyle O(n^{4})}"> operations, but that does not take <a href="/wiki/Combinatorial_explosion" title="Combinatorial explosion">combinatorial explosion</a> into account. </li> </ol></div></div> <div class="mw-heading mw-heading3"><h3 id="Citations">Citations</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=50" title="Edit section: Citations">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239543626"><div class="reflist reflist-columns references-column-width" style="column-width: 30em;"> <ol class="references"> <li id="cite_note-FOOTNOTEBurdenFaires1993401-1"><a href="#cite_ref-FOOTNOTEBurdenFaires1993401_1-0">^</a> <a href="#CITEREFBurdenFaires1993">Burden & Faires 1993</a>, p. 401. </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFGilbert_Strang" class="citation book cs1">Gilbert Strang. 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Linear Algebra and Its Applications. 692: 91–134. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.laa.2024.03.035">10.1016/j.laa.2024.03.035</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Linear+Algebra+and+Its+Applications&rft.atitle=Eigenvector+components+of+symmetric%2C+graph-related+matrices&rft.volume=692&rft.pages=91-134&rft.date=2024&rft_id=info%3Adoi%2F10.1016%2Fj.laa.2024.03.035&rft.aulast=Van+Mieghem&rft.aufirst=P.&rft_id=https%3A%2F%2Fdoi.org%2F10.1016%252Fj.laa.2024.03.035&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEigenvalues+and+eigenvectors" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=52" title="Edit section: Further reading">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239549316"><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubvan_der_Vorst2000" class="citation cs2">Golub, Gene F.; van der Vorst, Henk A. (2000), <a rel="nofollow" class="external text" href="https://dspace.library.uu.nl/bitstream/1874/2663/1/eighistory.pdf">"Eigenvalue Computation in the 20th Century"</a> (PDF), Journal of Computational and Applied Mathematics, 123 (1–2): 35–65, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000JCoAM.123...35G">2000JCoAM.123...35G</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0377-0427%2800%2900413-1">10.1016/S0377-0427(00)00413-1</a>, <a href="/wiki/Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1874%2F2663">1874/2663</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Computational+and+Applied+Mathematics&rft.atitle=Eigenvalue+Computation+in+the+20th+Century&rft.volume=123&rft.issue=1%E2%80%932&rft.pages=35-65&rft.date=2000&rft_id=info%3Ahdl%2F1874%2F2663&rft_id=info%3Adoi%2F10.1016%2FS0377-0427%2800%2900413-1&rft_id=info%3Abibcode%2F2000JCoAM.123...35G&rft.aulast=Golub&rft.aufirst=Gene+F.&rft.au=van+der+Vorst%2C+Henk+A.&rft_id=https%3A%2F%2Fdspace.library.uu.nl%2Fbitstream%2F1874%2F2663%2F1%2Feighistory.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEigenvalues+and+eigenvectors" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1">Hill, Roger (2009). <a rel="nofollow" class="external text" href="https://sixtysymbols.com/videos/eigenvalues.htm">"λ – Eigenvalues"</a>. Sixty Symbols. <a href="/wiki/Brady_Haran" title="Brady Haran">Brady Haran</a> for the <a href="/wiki/University_of_Nottingham" title="University of Nottingham">University of Nottingham</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Sixty+Symbols&rft.atitle=%CE%BB+%E2%80%93+Eigenvalues&rft.date=2009&rft.aulast=Hill&rft.aufirst=Roger&rft_id=https%3A%2F%2Fsixtysymbols.com%2Fvideos%2Feigenvalues.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEigenvalues+and+eigenvectors" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKuttler2017" class="citation cs2">Kuttler, Kenneth (2017), <a rel="nofollow" class="external text" href="https://math.byu.edu/~klkuttle/Linearalgebra.pdf">An introduction to linear algebra</a> (PDF), Brigham Young University</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+linear+algebra&rft.pub=Brigham+Young+University&rft.date=2017&rft.aulast=Kuttler&rft.aufirst=Kenneth&rft_id=https%3A%2F%2Fmath.byu.edu%2F~klkuttle%2FLinearalgebra.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEigenvalues+and+eigenvectors" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang1993" class="citation cs2">Strang, Gilbert (1993), Introduction to linear algebra, Wellesley, MA: Wellesley-Cambridge Press, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-9614088-5-5" title="Special:BookSources/0-9614088-5-5"><bdi>0-9614088-5-5</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+linear+algebra&rft.place=Wellesley%2C+MA&rft.pub=Wellesley-Cambridge+Press&rft.date=1993&rft.isbn=0-9614088-5-5&rft.aulast=Strang&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEigenvalues+and+eigenvectors" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrang2006" class="citation cs2">Strang, Gilbert (2006), Linear algebra and its applications, Belmont, CA: Thomson, Brooks/Cole, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-03-010567-6" title="Special:BookSources/0-03-010567-6"><bdi>0-03-010567-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+algebra+and+its+applications&rft.place=Belmont%2C+CA&rft.pub=Thomson%2C+Brooks%2FCole&rft.date=2006&rft.isbn=0-03-010567-6&rft.aulast=Strang&rft.aufirst=Gilbert&rfr_id=info%3Asid%2Fen.wikipedia.org%3AEigenvalues+and+eigenvectors" class="Z3988"></li></ul> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=53" title="Edit section: External links">edit</a>]</div> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1251242444"><table class="box-External_links plainlinks metadata ambox ambox-style ambox-external_links" role="presentation"><tbody><tr><td class="mbox-image"><div class="mbox-image-div"><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/40px-Edit-clear.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/60px-Edit-clear.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/f/f2/Edit-clear.svg/80px-Edit-clear.svg.png 2x" data-file-width="48" data-file-height="48" /></div></td><td class="mbox-text"><div class="mbox-text-span">This article's use of <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">external links</a> may not follow Wikipedia's policies or guidelines. Please <a class="external text" href="https://en.wikipedia.org/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit">improve this article</a> by removing <a href="/wiki/Wikipedia:What_Wikipedia_is_not#Wikipedia_is_not_a_mirror_or_a_repository_of_links,_images,_or_media_files" title="Wikipedia:What Wikipedia is not">excessive</a> or <a href="/wiki/Wikipedia:External_links" title="Wikipedia:External links">inappropriate</a> external links, and converting useful links where appropriate into <a href="/wiki/Wikipedia:Citing_sources" title="Wikipedia:Citing sources">footnote references</a>. (December 2019) (<a href="/wiki/Help:Maintenance_template_removal" title="Help:Maintenance template removal">Learn how and when to remove this message</a>)</div></td></tr></tbody></table> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png" decoding="async" width="40" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/60px-Wikibooks-logo-en-noslogan.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/80px-Wikibooks-logo-en-noslogan.svg.png 2x" data-file-width="400" data-file-height="400" /></div> <div class="side-box-text plainlist">The Wikibook <a href="https://en.wikibooks.org/wiki/Linear_Algebra" class="extiw" title="wikibooks:Linear Algebra">Linear Algebra</a> has a page on the topic of: <a href="https://en.wikibooks.org/wiki/Linear_Algebra/Eigenvalues_and_Eigenvectors" class="extiw" title="wikibooks:Linear Algebra/Eigenvalues and Eigenvectors">Eigenvalues and Eigenvectors</a></div></div> </div> <ul><li><a rel="nofollow" class="external text" href="https://physlink.com/education/AskExperts/ae520.cfm">What are Eigen Values?</a> – non-technical introduction from PhysLink.com's "Ask the Experts"</li> <li><a rel="nofollow" class="external text" href="https://people.revoledu.com/kardi/tutorial/LinearAlgebra/EigenValueEigenVector.html">Eigen Values and Eigen Vectors Numerical Examples</a> – Tutorial and Interactive Program from Revoledu.</li> <li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20100325112901/https://khanexercises.appspot.com/video?v=PhfbEr2btGQ">Introduction to Eigen Vectors and Eigen Values</a> – lecture from Khan Academy</li> <li><a rel="nofollow" class="external text" href="https://youtube.com/watch?v=PFDu9oVAE-g&list=PLZHQObOWTQDPD3MizzM2xVFitgF8hE_ab&index=14">Eigenvectors and eigenvalues | Essence of linear algebra, chapter 10</a> – A visual explanation with <a href="/wiki/3Blue1Brown" title="3Blue1Brown">3Blue1Brown</a></li> <li><a rel="nofollow" class="external text" href="https://symbolab.com/solver/matrix-eigenvectors-calculator">Matrix Eigenvectors Calculator</a> from Symbolab (Click on the bottom right button of the 2×12 grid to select a matrix size. Select an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n\times n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> <mo>×</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n\times n}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59d2b4cb72e304526cf5b5887147729ea259da78" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.63ex; height:1.676ex;" alt="{\displaystyle n\times n}"> size (for a square matrix), then fill out the entries numerically and click on the Go button. It can accept complex numbers as well.)</li></ul> <a href="/wiki/File:Wikiversity_logo_2017.svg" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/16px-Wikiversity_logo_2017.svg.png" decoding="async" width="16" height="13" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/24px-Wikiversity_logo_2017.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/32px-Wikiversity_logo_2017.svg.png 2x" data-file-width="626" data-file-height="512" /></a> Wikiversity uses introductory physics to introduce <a href="https://en.wikiversity.org/wiki/Physics/A/Eigenvalues_for_beginners" class="extiw" title="v:Physics/A/Eigenvalues for beginners">Eigenvalues and eigenvectors</a> <div class="mw-heading mw-heading3"><h3 id="Theory">Theory</h3>[<a href="/w/index.php?title=Eigenvalues_and_eigenvectors&action=edit&section=54" title="Edit section: Theory">edit</a>]</div> <ul><li><a rel="nofollow" class="external text" href="https://sosmath.com/matrix/eigen1/eigen1.html">Computation of Eigenvalues</a></li> <li><a rel="nofollow" class="external text" href="https://cs.utk.edu/~dongarra/etemplates/index.html">Numerical solution of eigenvalue problems</a> Edited by Zhaojun Bai, <a href="/wiki/James_Demmel" title="James Demmel">James Demmel</a>, Jack Dongarra, Axel Ruhe, and <a href="/wiki/Henk_van_der_Vorst" title="Henk van der Vorst">Henk van der Vorst</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl 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title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Linear_algebra" title="Special:EditPage/Template:Linear algebra"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Linear_algebra" style="font-size:114%;margin:0 4em"><a href="/wiki/Linear_algebra" title="Linear algebra">Linear algebra</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><a href="/wiki/Outline_of_linear_algebra" title="Outline of linear algebra">Outline</a></li> <li><a href="/wiki/Glossary_of_linear_algebra" title="Glossary of linear algebra">Glossary</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Basic concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Scalar_(mathematics)" title="Scalar (mathematics)">Scalar</a></li> <li><a href="/wiki/Euclidean_vector" title="Euclidean vector">Vector</a></li> <li><a href="/wiki/Vector_space" title="Vector space">Vector space</a></li> <li><a href="/wiki/Scalar_multiplication" title="Scalar multiplication">Scalar multiplication</a></li> <li><a href="/wiki/Vector_projection" title="Vector projection">Vector projection</a></li> <li><a href="/wiki/Linear_span" title="Linear span">Linear span</a></li> <li><a href="/wiki/Linear_map" title="Linear map">Linear map</a></li> <li><a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Linear projection</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Linear_combination" title="Linear combination">Linear combination</a></li> <li><a href="/wiki/Multilinear_map" title="Multilinear map">Multilinear map</a></li> <li><a href="/wiki/Basis_(linear_algebra)" title="Basis (linear algebra)">Basis</a></li> <li><a href="/wiki/Change_of_basis" title="Change of basis">Change of basis</a></li> <li><a href="/wiki/Row_and_column_vectors" title="Row and column vectors">Row and column vectors</a></li> <li><a href="/wiki/Row_and_column_spaces" title="Row and column spaces">Row and column spaces</a></li> <li><a href="/wiki/Kernel_(linear_algebra)" title="Kernel (linear algebra)">Kernel</a></li> <li><a class="mw-selflink selflink">Eigenvalues and eigenvectors</a></li> <li><a href="/wiki/Transpose" title="Transpose">Transpose</a></li> <li><a href="/wiki/System_of_linear_equations" title="System of linear equations">Linear equations</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="6" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/Euclidean_space" title="Euclidean space"><img alt="Three dimensional Euclidean space" src="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/80px-Linear_subspaces_with_shading.svg.png" decoding="async" width="80" height="58" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/120px-Linear_subspaces_with_shading.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/2f/Linear_subspaces_with_shading.svg/160px-Linear_subspaces_with_shading.svg.png 2x" data-file-width="325" data-file-height="236" /></a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrices</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Matrix_decomposition" title="Matrix decomposition">Decomposition</a></li> <li><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li> <li><a href="/wiki/Minor_(linear_algebra)" title="Minor (linear algebra)">Minor</a></li> <li><a href="/wiki/Matrix_multiplication" title="Matrix multiplication">Multiplication</a></li> <li><a href="/wiki/Rank_(linear_algebra)" title="Rank (linear algebra)">Rank</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li> <li><a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a></li> <li><a href="/wiki/Gaussian_elimination" title="Gaussian elimination">Gaussian elimination</a></li> <li><a href="/wiki/Productive_matrix" title="Productive matrix">Productive matrix</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Bilinear_map" title="Bilinear map">Bilinear</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Orthogonality" title="Orthogonality">Orthogonality</a></li> <li><a href="/wiki/Dot_product" title="Dot product">Dot product</a></li> <li><a href="/wiki/Hadamard_product_(matrices)" title="Hadamard product (matrices)">Hadamard product</a></li> <li><a href="/wiki/Inner_product_space" title="Inner product space">Inner product space</a></li> <li><a href="/wiki/Outer_product" title="Outer product">Outer product</a></li> <li><a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a></li> <li><a href="/wiki/Gram%E2%80%93Schmidt_process" title="Gram–Schmidt process">Gram–Schmidt process</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Determinant" title="Determinant">Determinant</a></li> <li><a href="/wiki/Cross_product" title="Cross product">Cross product</a></li> <li><a href="/wiki/Triple_product" title="Triple product">Triple product</a></li> <li><a href="/wiki/Seven-dimensional_cross_product" title="Seven-dimensional cross product">Seven-dimensional cross product</a></li> <li><a href="/wiki/Geometric_algebra" title="Geometric algebra">Geometric algebra</a></li> <li><a href="/wiki/Exterior_algebra" title="Exterior algebra">Exterior algebra</a></li> <li><a href="/wiki/Bivector" title="Bivector">Bivector</a></li> <li><a href="/wiki/Multivector" title="Multivector">Multivector</a></li> <li><a href="/wiki/Tensor" title="Tensor">Tensor</a></li> <li><a href="/wiki/Outermorphism" title="Outermorphism">Outermorphism</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Vector_space" title="Vector space">Vector space</a> constructions</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Dual_space" title="Dual space">Dual</a></li> <li><a href="/wiki/Direct_sum_of_modules#Construction_for_two_vector_spaces" title="Direct sum of modules">Direct sum</a></li> <li><a href="/wiki/Function_space#In_linear_algebra" title="Function space">Function space</a></li> <li><a href="/wiki/Quotient_space_(linear_algebra)" title="Quotient space (linear algebra)">Quotient</a></li> <li><a href="/wiki/Linear_subspace" title="Linear subspace">Subspace</a></li> <li><a href="/wiki/Tensor_product" title="Tensor product">Tensor product</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Numerical_linear_algebra" title="Numerical linear algebra">Numerical</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Floating-point_arithmetic" title="Floating-point arithmetic">Floating-point</a></li> <li><a href="/wiki/Numerical_stability" title="Numerical stability">Numerical stability</a></li> <li><a href="/wiki/Basic_Linear_Algebra_Subprograms" title="Basic Linear Algebra Subprograms">Basic Linear Algebra Subprograms</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse matrix</a></li> <li><a href="/wiki/Comparison_of_linear_algebra_libraries" title="Comparison of linear algebra libraries">Comparison of linear algebra libraries</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Linear_algebra" title="Category:Linear algebra">Category</a></li></ul> </div></td></tr></tbody></table></div> <div class="navbox-styles"><link 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style="width:1%"><a href="/wiki/Foundations_of_mathematics" title="Foundations of mathematics">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Category_theory" title="Category theory">Category theory</a></li> <li><a href="/wiki/Information_theory" title="Information theory">Information theory</a></li> <li><a href="/wiki/Mathematical_logic" title="Mathematical logic">Mathematical logic</a></li> <li><a href="/wiki/Philosophy_of_mathematics" title="Philosophy of mathematics">Philosophy of mathematics</a></li> <li><a href="/wiki/Set_theory" title="Set theory">Set theory</a></li> <li><a href="/wiki/Type_theory" title="Type theory">Type theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Algebra" title="Algebra">Algebra</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract</a></li> <li><a href="/wiki/Commutative_algebra" title="Commutative algebra">Commutative</a></li> <li><a href="/wiki/Elementary_algebra" title="Elementary algebra">Elementary</a></li> <li><a href="/wiki/Group_theory" title="Group theory">Group theory</a></li> <li><a href="/wiki/Linear_algebra" title="Linear algebra">Linear</a></li> <li><a href="/wiki/Multilinear_algebra" title="Multilinear algebra">Multilinear</a></li> <li><a href="/wiki/Universal_algebra" title="Universal algebra">Universal</a></li> <li><a href="/wiki/Homological_algebra" title="Homological algebra">Homological</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Mathematical_analysis" title="Mathematical analysis">Analysis</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a 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<ul><li><a href="/wiki/Combinatorics" title="Combinatorics">Combinatorics</a></li> <li><a href="/wiki/Graph_theory" title="Graph theory">Graph theory</a></li> <li><a href="/wiki/Order_theory" title="Order theory">Order theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geometry" title="Geometry">Geometry</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Algebraic_geometry" title="Algebraic geometry">Algebraic</a></li> <li><a href="/wiki/Analytic_geometry" title="Analytic geometry">Analytic</a></li> <li><a href="/wiki/Arithmetic_geometry" title="Arithmetic geometry">Arithmetic</a></li> <li><a href="/wiki/Differential_geometry" title="Differential geometry">Differential</a></li> <li><a href="/wiki/Discrete_geometry" title="Discrete geometry">Discrete</a></li> <li><a href="/wiki/Euclidean_geometry" title="Euclidean geometry">Euclidean</a></li> <li><a href="/wiki/Finite_geometry" title="Finite geometry">Finite</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Number_theory" title="Number theory">Number theory</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Arithmetic" title="Arithmetic">Arithmetic</a></li> <li><a href="/wiki/Algebraic_number_theory" title="Algebraic number theory">Algebraic number theory</a></li> <li><a href="/wiki/Analytic_number_theory" title="Analytic number theory">Analytic number theory</a></li> <li><a href="/wiki/Diophantine_geometry" title="Diophantine geometry">Diophantine geometry</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Topology" title="Topology">Topology</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/General_topology" title="General topology">General</a></li> <li><a href="/wiki/Algebraic_topology" title="Algebraic topology">Algebraic</a></li> <li><a href="/wiki/Differential_topology" title="Differential topology">Differential</a></li> <li><a href="/wiki/Geometric_topology" title="Geometric topology">Geometric</a></li> <li><a href="/wiki/Homotopy_theory" title="Homotopy theory">Homotopy theory</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Applied_mathematics" title="Applied mathematics">Applied</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Engineering_mathematics" title="Engineering mathematics">Engineering mathematics</a></li> <li><a href="/wiki/Mathematical_and_theoretical_biology" title="Mathematical and theoretical biology">Mathematical biology</a></li> <li><a href="/wiki/Mathematical_chemistry" title="Mathematical chemistry">Mathematical chemistry</a></li> <li><a href="/wiki/Mathematical_economics" title="Mathematical economics">Mathematical economics</a></li> <li><a href="/wiki/Mathematical_finance" title="Mathematical finance">Mathematical finance</a></li> <li><a href="/wiki/Mathematical_physics" title="Mathematical physics">Mathematical physics</a></li> <li><a href="/wiki/Mathematical_psychology" title="Mathematical psychology">Mathematical psychology</a></li> <li><a href="/wiki/Mathematical_sociology" title="Mathematical sociology">Mathematical sociology</a></li> <li><a href="/wiki/Mathematical_statistics" title="Mathematical statistics">Mathematical statistics</a></li> <li><a href="/wiki/Probability_theory" title="Probability theory">Probability</a></li> <li><a href="/wiki/Statistics" title="Statistics">Statistics</a></li> <li><a href="/wiki/Systems_science" title="Systems science">Systems science</a> <ul><li><a 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title="Recreational mathematics">Recreational mathematics</a></li> <li><a href="/wiki/Mathematics_and_art" title="Mathematics and art">Mathematics and art</a></li> <li><a href="/wiki/Mathematics_education" title="Mathematics education">Mathematics education</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" 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Eigenvalues and eigenvectors - Wikipedia