Gram–Schmidt process

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href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" data-event-name="menu.edit" data-mw="interface" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet edit-page menu__item--page-actions-edit"> Edit </a></li> </ul> </nav> <div id="mw-content-subtitle"></div> </div> <div id="bodyContent" class="content"> <div id="mw-content-text" class="mw-body-content"> <script>function mfTempOpenSection(id){var block=document.getElementById("mf-section-"+id);block.className+=" open-block";block.previousSibling.className+=" open-block";}</script> <div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"> <section class="mf-section-0" id="mf-section-0"> In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematics">mathematics</a>, particularly <a href="https://en-m-wikipedia-org.translate.goog/wiki/Linear_algebra?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Linear algebra">linear algebra</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Numerical_analysis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Numerical analysis">numerical analysis</a>, the Gram–Schmidt process or Gram-Schmidt algorithm is a way of finding a set of two or more vectors that are perpendicular to each other. <figure class="mw-halign-right" typeof="mw:File/Frame"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Gram%E2%80%93Schmidt_process.svg?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/9/97/Gram%E2%80%93Schmidt_process.svg/350px-Gram%E2%80%93Schmidt_process.svg.png" decoding="async" width="350" height="180" class="mw-file-element" srcset="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/9/97/Gram%25E2%2580%2593Schmidt_process.svg/525px-Gram%25E2%2580%2593Schmidt_process.svg.png 1.5x,https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://upload.wikimedia.org/wikipedia/commons/thumb/9/97/Gram%25E2%2580%2593Schmidt_process.svg/700px-Gram%25E2%2580%2593Schmidt_process.svg.png 2x" data-file-width="350" data-file-height="180"></a> <figcaption> The first two steps of the Gram–Schmidt process </figcaption> </figure> By technical definition, it is a method of constructing an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthonormal_basis?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthonormal basis">orthonormal basis</a> from a set of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Vector_(geometry)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Vector (geometry)">vectors</a> in an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Inner_product_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Inner product space">inner product space</a>, most commonly the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Euclidean_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Euclidean space">Euclidean space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{n}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" 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 {R} ^{n}}"> equipped with the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Standard_inner_product?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Standard inner product">standard inner product</a>. The Gram–Schmidt process takes a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Finite_set?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Finite set">finite</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Linearly_independent?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Linearly independent">linearly independent</a> set of vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{k}\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mo> = </mo> <mo fence="false" stretchy="false"> { </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{k}\}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a05cfe141bb15684e26bb802a9582f2648d32c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.066ex; height:2.843ex;" alt="{\displaystyle S=\{\mathbf {v} _{1},\ldots ,\mathbf {v} _{k}\}}"> for k ≤ n and generates an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_set?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Orthogonal set">orthogonal set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S'=\{\mathbf {u} _{1},\ldots ,\mathbf {u} _{k}\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> S </mi> <mo> ′ </mo> </msup> <mo> = </mo> <mo fence="false" stretchy="false"> { </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S'=\{\mathbf {u} _{1},\ldots ,\mathbf {u} _{k}\}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6de4b1d1088fe7f22c9135d7224530345c32b47d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.922ex; height:3.009ex;" alt="{\displaystyle S'=\{\mathbf {u} _{1},\ldots ,\mathbf {u} _{k}\}}"> that spans the same <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}">-dimensional subspace of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{n}} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d" 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 {R} ^{n}}"> as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S} </annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4611d85173cd3b508e67077d4a1252c9c05abca2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}">. The method is named after <a href="https://en-m-wikipedia-org.translate.goog/wiki/J%C3%B8rgen_Pedersen_Gram?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Jørgen Pedersen Gram">Jørgen Pedersen Gram</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Erhard_Schmidt?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Erhard Schmidt">Erhard Schmidt</a>, but <a href="https://en-m-wikipedia-org.translate.goog/wiki/Pierre-Simon_Laplace?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Pierre-Simon Laplace">Pierre-Simon Laplace</a> had been familiar with it before Gram and Schmidt.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-1">[1]</a> In the theory of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Lie_group_decompositions?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Lie group decompositions">Lie group decompositions</a>, it is generalized by the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Iwasawa_decomposition?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Iwasawa decomposition">Iwasawa decomposition</a>. The application of the Gram–Schmidt process to the column vectors of a full column <a href="https://en-m-wikipedia-org.translate.goog/wiki/Rank_(linear_algebra)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Rank (linear algebra)">rank</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Matrix_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Matrix (mathematics)">matrix</a> yields the <a href="https://en-m-wikipedia-org.translate.goog/wiki/QR_decomposition?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="QR decomposition">QR decomposition</a> (it is decomposed into an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthogonal matrix">orthogonal</a> and a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Triangular_matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Triangular matrix">triangular matrix</a>). <div id="toc" class="toc" role="navigation" aria-labelledby="mw-toc-heading"> <input type="checkbox" role="button" id="toctogglecheckbox" class="toctogglecheckbox" style="display:none"> <div class="toctitle" lang="en" dir="ltr"> <h2 id="mw-toc-heading">Contents</h2><label class="toctogglelabel" for="toctogglecheckbox"></label> </div> <ul> <li class="toclevel-1 tocsection-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#The_Gram%E2%80%93Schmidt_process">1 The Gram–Schmidt process</a></li> <li class="toclevel-1 tocsection-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Example">2 Example</a> <ul> <li class="toclevel-2 tocsection-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Euclidean_space">2.1 Euclidean space</a></li> </ul></li> <li class="toclevel-1 tocsection-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Properties">3 Properties</a></li> <li class="toclevel-1 tocsection-5"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Numerical_stability">4 Numerical stability</a></li> <li class="toclevel-1 tocsection-6"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Algorithm">5 Algorithm</a></li> <li class="toclevel-1 tocsection-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Via_Gaussian_elimination">6 Via Gaussian elimination</a></li> <li class="toclevel-1 tocsection-8"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Determinant_formula">7 Determinant formula</a></li> <li class="toclevel-1 tocsection-9"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Expressed_using_geometric_algebra">8 Expressed using geometric algebra</a></li> <li class="toclevel-1 tocsection-10"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Alternatives">9 Alternatives</a></li> <li class="toclevel-1 tocsection-11"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Run-time_complexity">10 Run-time complexity</a></li> <li class="toclevel-1 tocsection-12"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#See_also">11 See also</a></li> <li class="toclevel-1 tocsection-13"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#References">12 References</a></li> <li class="toclevel-1 tocsection-14"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Notes">13 Notes</a></li> <li class="toclevel-1 tocsection-15"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Sources">14 Sources</a></li> <li class="toclevel-1 tocsection-16"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#External_links">15 External links</a></li> </ul> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(1)"> <h2 id="The_Gram–Schmidt_process">The Gram–Schmidt process</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=1&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: The Gram–Schmidt process" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-1 collapsible-block" id="mf-section-1"> <figure typeof="mw:File/Thumb"> <a href="https://en-m-wikipedia-org.translate.goog/wiki/File:Gram-Schmidt_orthonormalization_process.gif?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-file-description"> <noscript> <img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Gram-Schmidt_orthonormalization_process.gif/400px-Gram-Schmidt_orthonormalization_process.gif" decoding="async" width="400" height="300" class="mw-file-element" data-file-width="440" data-file-height="330"> </noscript><span class="lazy-image-placeholder" style="width: 400px;height: 300px;" data-src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Gram-Schmidt_orthonormalization_process.gif/400px-Gram-Schmidt_orthonormalization_process.gif" data-width="400" data-height="300" data-srcset="//upload.wikimedia.org/wikipedia/commons/e/ee/Gram-Schmidt_orthonormalization_process.gif 1.5x" data-class="mw-file-element"> </a> <figcaption> The modified Gram-Schmidt process being executed on three linearly independent, non-orthogonal vectors of a basis for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{3}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.732ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f936ddf584f8f3dd2a0ed08917001b7a404c10b5" data-alt="{\displaystyle \mathbb {R} ^{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Click on image for details. Modification is explained in the Numerical Stability section of this article. </figcaption> </figure> The <a href="https://en-m-wikipedia-org.translate.goog/wiki/Vector_projection?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Vector projection">vector projection</a> of a 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> <noscript> <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} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" data-alt="{\displaystyle \mathbf {v} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  on a nonzero vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" data-alt="{\displaystyle \mathbf {u} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is defined as<a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-2">[note 1]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {v} ,\mathbf {u} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} ,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false"> ⟨ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo fence="false" stretchy="false"> ⟩ </mo> </mrow> <mrow> <mo fence="false" stretchy="false"> ⟨ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo fence="false" stretchy="false"> ⟩ </mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {v} ,\mathbf {u} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea472b7ba11e1fd1749afaaf007ef44519eb2f8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.849ex; height:6.509ex;" alt="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {v} ,\mathbf {u} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} ,}"> </noscript><span class="lazy-image-placeholder" style="width: 20.849ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9ea472b7ba11e1fd1749afaaf007ef44519eb2f8" data-alt="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {v} ,\mathbf {u} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} ,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo fence="false" stretchy="false"> ⟩ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121be2362b7f59581dc4779932a8e3c71240f9e3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.74ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle }"> </noscript><span class="lazy-image-placeholder" style="width: 5.74ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/121be2362b7f59581dc4779932a8e3c71240f9e3" data-alt="{\displaystyle \langle \mathbf {v} ,\mathbf {u} \rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  denotes the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Inner_product?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Inner product">inner product</a> of the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" data-alt="{\displaystyle \mathbf {u} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <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> <noscript> <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} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" data-alt="{\displaystyle \mathbf {v} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This means that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f6b492f8f5213ce706a9aaf7a608548b6af115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.581ex; height:2.843ex;" alt="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )}"> </noscript><span class="lazy-image-placeholder" style="width: 8.581ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f6b492f8f5213ce706a9aaf7a608548b6af115" data-alt="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonal_projection?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Orthogonal projection">orthogonal projection</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> <noscript> <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} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.411ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/35c1866e359fbfd2e0f606c725ba5cc37a5195d6" data-alt="{\displaystyle \mathbf {v} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  onto the line spanned by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" data-alt="{\displaystyle \mathbf {u} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . If <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.485ex; height:1.676ex;" alt="{\displaystyle \mathbf {u} }"> </noscript><span class="lazy-image-placeholder" style="width: 1.485ex;height: 1.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/261e20fe101de02a771021d9d4466c0ad3e352d7" data-alt="{\displaystyle \mathbf {u} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the zero vector, then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f6b492f8f5213ce706a9aaf7a608548b6af115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.581ex; height:2.843ex;" alt="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )}"> </noscript><span class="lazy-image-placeholder" style="width: 8.581ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5f6b492f8f5213ce706a9aaf7a608548b6af115" data-alt="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is defined as the zero vector. Given <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{k}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916a7303fbee0f437f7214458a0dd6de886fb53a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.143ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.143ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916a7303fbee0f437f7214458a0dd6de886fb53a" data-alt="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  the Gram–Schmidt process defines the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbb786f13046e085458c34a87f3951f1808b8ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.292ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.292ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbb786f13046e085458c34a87f3951f1808b8ca" data-alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as follows: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {u} _{1}&=\mathbf {v} _{1},&\!\mathbf {e} _{1}&={\frac {\mathbf {u} _{1}}{\|\mathbf {u} _{1}\|}}\\\mathbf {u} _{2}&=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2}),&\!\mathbf {e} _{2}&={\frac {\mathbf {u} _{2}}{\|\mathbf {u} _{2}\|}}\\\mathbf {u} _{3}&=\mathbf {v} _{3}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{3})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{3}),&\!\mathbf {e} _{3}&={\frac {\mathbf {u} _{3}}{\|\mathbf {u} _{3}\|}}\\\mathbf {u} _{4}&=\mathbf {v} _{4}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{3}}(\mathbf {v} _{4}),&\!\mathbf {e} _{4}&={\mathbf {u} _{4} \over \|\mathbf {u} _{4}\|}\\&{}\ \ \vdots &&{}\ \ \vdots \\\mathbf {u} _{k}&=\mathbf {v} _{k}-\sum _{j=1}^{k-1}\operatorname {proj} _{\mathbf {u} _{j}}(\mathbf {v} _{k}),&\!\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}}{\|\mathbf {u} _{k}\|}}.\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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mrow> <mo fence="false" stretchy="false"> ‖ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ‖ </mo> </mrow> </mfrac> </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> <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> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mrow> <mo fence="false" stretchy="false"> ‖ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ‖ </mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mrow> <mo fence="false" stretchy="false"> ‖ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ‖ </mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <mrow> <mo fence="false" stretchy="false"> ‖ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ‖ </mo> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mtext>   </mtext> <mtext>   </mtext> <mo> ⋮ </mo> </mtd> <mtd></mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> </mrow> <mtext>   </mtext> <mtext>   </mtext> <mo> ⋮ </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </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> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> </mtd> <mtd> <mspace width="negativethinmathspace"></mspace> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mrow> <mo fence="false" stretchy="false"> ‖ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo fence="false" stretchy="false"> ‖ </mo> </mrow> </mfrac> </mrow> <mo> . </mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\mathbf {u} _{1}&=\mathbf {v} _{1},&\!\mathbf {e} _{1}&={\frac {\mathbf {u} _{1}}{\|\mathbf {u} _{1}\|}}\\\mathbf {u} _{2}&=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2}),&\!\mathbf {e} _{2}&={\frac {\mathbf {u} _{2}}{\|\mathbf {u} _{2}\|}}\\\mathbf {u} _{3}&=\mathbf {v} _{3}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{3})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{3}),&\!\mathbf {e} _{3}&={\frac {\mathbf {u} _{3}}{\|\mathbf {u} _{3}\|}}\\\mathbf {u} _{4}&=\mathbf {v} _{4}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{3}}(\mathbf {v} _{4}),&\!\mathbf {e} _{4}&={\mathbf {u} _{4} \over \|\mathbf {u} _{4}\|}\\&{}\ \ \vdots &&{}\ \ \vdots \\\mathbf {u} _{k}&=\mathbf {v} _{k}-\sum _{j=1}^{k-1}\operatorname {proj} _{\mathbf {u} _{j}}(\mathbf {v} _{k}),&\!\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}}{\|\mathbf {u} _{k}\|}}.\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad89bad7c5fb0df82786c5b6938dce503af2dd0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.671ex; width:65.511ex; height:34.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {u} _{1}&=\mathbf {v} _{1},&\!\mathbf {e} _{1}&={\frac {\mathbf {u} _{1}}{\|\mathbf {u} _{1}\|}}\\\mathbf {u} _{2}&=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2}),&\!\mathbf {e} _{2}&={\frac {\mathbf {u} _{2}}{\|\mathbf {u} _{2}\|}}\\\mathbf {u} _{3}&=\mathbf {v} _{3}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{3})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{3}),&\!\mathbf {e} _{3}&={\frac {\mathbf {u} _{3}}{\|\mathbf {u} _{3}\|}}\\\mathbf {u} _{4}&=\mathbf {v} _{4}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{3}}(\mathbf {v} _{4}),&\!\mathbf {e} _{4}&={\mathbf {u} _{4} \over \|\mathbf {u} _{4}\|}\\&{}\ \ \vdots &&{}\ \ \vdots \\\mathbf {u} _{k}&=\mathbf {v} _{k}-\sum _{j=1}^{k-1}\operatorname {proj} _{\mathbf {u} _{j}}(\mathbf {v} _{k}),&\!\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}}{\|\mathbf {u} _{k}\|}}.\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 65.511ex;height: 34.509ex;vertical-align: -16.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ad89bad7c5fb0df82786c5b6938dce503af2dd0" data-alt="{\displaystyle {\begin{aligned}\mathbf {u} _{1}&=\mathbf {v} _{1},&\!\mathbf {e} _{1}&={\frac {\mathbf {u} _{1}}{\|\mathbf {u} _{1}\|}}\\\mathbf {u} _{2}&=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2}),&\!\mathbf {e} _{2}&={\frac {\mathbf {u} _{2}}{\|\mathbf {u} _{2}\|}}\\\mathbf {u} _{3}&=\mathbf {v} _{3}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{3})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{3}),&\!\mathbf {e} _{3}&={\frac {\mathbf {u} _{3}}{\|\mathbf {u} _{3}\|}}\\\mathbf {u} _{4}&=\mathbf {v} _{4}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{4})-\operatorname {proj} _{\mathbf {u} _{3}}(\mathbf {v} _{4}),&\!\mathbf {e} _{4}&={\mathbf {u} _{4} \over \|\mathbf {u} _{4}\|}\\&{}\ \ \vdots &&{}\ \ \vdots \\\mathbf {u} _{k}&=\mathbf {v} _{k}-\sum _{j=1}^{k-1}\operatorname {proj} _{\mathbf {u} _{j}}(\mathbf {v} _{k}),&\!\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}}{\|\mathbf {u} _{k}\|}}.\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbb786f13046e085458c34a87f3951f1808b8ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.292ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.292ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbb786f13046e085458c34a87f3951f1808b8ca" data-alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the required system of orthogonal vectors, and the normalized vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5774fb599a7237f1e74e1f953e93c157bff95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.772ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.772ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5774fb599a7237f1e74e1f953e93c157bff95c" data-alt="{\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  form an <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthonormal_set?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Orthonormal set">orthonormal set</a>. The calculation of the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbb786f13046e085458c34a87f3951f1808b8ca" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.292ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.292ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9bbb786f13046e085458c34a87f3951f1808b8ca" data-alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is known as Gram–Schmidt <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonalization?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthogonalization">orthogonalization</a>, and the calculation of the sequence <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5774fb599a7237f1e74e1f953e93c157bff95c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.772ex; height:2.009ex;" alt="{\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.772ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c5774fb599a7237f1e74e1f953e93c157bff95c" data-alt="{\displaystyle \mathbf {e} _{1},\ldots ,\mathbf {e} _{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is known as Gram–Schmidt <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthonormalization?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Orthonormalization">orthonormalization</a>. To check that these formulas yield an orthogonal sequence, first compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba0654d7f642b264c333ac18df0cff72db6615d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.922ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle }"> </noscript><span class="lazy-image-placeholder" style="width: 7.922ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5ba0654d7f642b264c333ac18df0cff72db6615d" data-alt="{\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  by substituting the above formula for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{2}}"><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> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636f3f24b87f95f1c772c4ac0b579c9ee7b43899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.54ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.54ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636f3f24b87f95f1c772c4ac0b579c9ee7b43899" data-alt="{\displaystyle \mathbf {u} _{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : we get zero. Then use this to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{3}\rangle }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{3}\rangle } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64ed9cf376aff4fc7573e494b33fa2f4cc71efc9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.922ex; height:2.843ex;" alt="{\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{3}\rangle }"> </noscript><span class="lazy-image-placeholder" style="width: 7.922ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/64ed9cf376aff4fc7573e494b33fa2f4cc71efc9" data-alt="{\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{3}\rangle }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  again by substituting the formula for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{3}}"><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> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8d2b326bb13caf8a0025fe53f4386793657eff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.54ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.54ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5b8d2b326bb13caf8a0025fe53f4386793657eff" data-alt="{\displaystyle \mathbf {u} _{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> : we get zero. For arbitrary <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.211ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" data-alt="{\displaystyle k}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  the proof is accomplished by <a href="https://en-m-wikipedia-org.translate.goog/wiki/Mathematical_induction?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Mathematical induction">mathematical induction</a>. Geometrically, this method proceeds as follows: to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{i}}"><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"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99336b5d39198e16034613b2c3cc54d898c3d24c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.285ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99336b5d39198e16034613b2c3cc54d898c3d24c" data-alt="{\displaystyle \mathbf {u} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , it projects <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}}"><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> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.211ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" data-alt="{\displaystyle \mathbf {v} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  orthogonally onto the subspace <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> </noscript><span class="lazy-image-placeholder" style="width: 1.783ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" data-alt="{\displaystyle U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{i-1}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{i-1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f5b835f50eab6d45267d33e5d3a5a21ea5114f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.103ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{i-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.103ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6f5b835f50eab6d45267d33e5d3a5a21ea5114f" data-alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{i-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , which is the same as the subspace generated by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-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> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13956a30f2b41cec722e4a8f64208955d07bb11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.955ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.955ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13956a30f2b41cec722e4a8f64208955d07bb11" data-alt="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The vector <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{i}}"><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"> <mi> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99336b5d39198e16034613b2c3cc54d898c3d24c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.285ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.285ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/99336b5d39198e16034613b2c3cc54d898c3d24c" data-alt="{\displaystyle \mathbf {u} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is then defined to be the difference between <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}}"><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> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.211ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" data-alt="{\displaystyle \mathbf {v} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and this projection, guaranteed to be orthogonal to all of the vectors in the subspace <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> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}"> </noscript><span class="lazy-image-placeholder" style="width: 1.783ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/458a728f53b9a0274f059cd695e067c430956025" data-alt="{\displaystyle U}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The Gram–Schmidt process also applies to a linearly independent <a href="https://en-m-wikipedia-org.translate.goog/wiki/Countably_infinite?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Countably infinite">countably infinite</a> sequence {vi}i. The result is an orthogonal (or orthonormal) sequence {ui}i such that for natural number n: the algebraic span of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{n}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{n}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae3cee159c711789d27ca7ee7653de05c45eb5a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.273ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.273ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ae3cee159c711789d27ca7ee7653de05c45eb5a" data-alt="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the same as that of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be69eacc0a24636857d567f9d32069dc0e488139" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.422ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.422ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/be69eacc0a24636857d567f9d32069dc0e488139" data-alt="{\displaystyle \mathbf {u} _{1},\ldots ,\mathbf {u} _{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . If the Gram–Schmidt process is applied to a linearly dependent sequence, it outputs the 0 vector on 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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 0.802ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/add78d8608ad86e54951b8c8bd6c8d8416533d20" data-alt="{\displaystyle i}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> th step, assuming that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{i}}"><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> i </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{i}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.211ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{i}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.211ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51747274b58895dd357bb270ba1b5cb71e4fa355" data-alt="{\displaystyle \mathbf {v} _{i}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a linear combination of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-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> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> i </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13956a30f2b41cec722e4a8f64208955d07bb11" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.955ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.955ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e13956a30f2b41cec722e4a8f64208955d07bb11" data-alt="{\displaystyle \mathbf {v} _{1},\ldots ,\mathbf {v} _{i-1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . If an orthonormal basis is to be produced, then the algorithm should test for zero vectors in the output and discard them because no multiple of a zero vector can have a length of 1. The number of vectors output by the algorithm will then be the dimension of the space spanned by the original inputs. A variant of the Gram–Schmidt process using <a href="https://en-m-wikipedia-org.translate.goog/wiki/Transfinite_recursion?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Transfinite recursion">transfinite recursion</a> applied to a (possibly uncountably) infinite sequence of vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{\alpha })_{\alpha <\lambda }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> </mrow> </msub> <msub> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> <mo> < </mo> <mi> λ </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (v_{\alpha })_{\alpha <\lambda }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb44bc2105d877ca44d2a4c75e35fe5b71acceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.742ex; height:2.843ex;" alt="{\displaystyle (v_{\alpha })_{\alpha <\lambda }}"> </noscript><span class="lazy-image-placeholder" style="width: 7.742ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb44bc2105d877ca44d2a4c75e35fe5b71acceb" data-alt="{\displaystyle (v_{\alpha })_{\alpha <\lambda }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  yields a set of orthonormal vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{\alpha })_{\alpha <\kappa }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> </mrow> </msub> <msub> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> <mo> < </mo> <mi> κ </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (u_{\alpha })_{\alpha <\kappa }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f7ba81b2efe577fa80b011abed68f41b7de827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.933ex; height:2.843ex;" alt="{\displaystyle (u_{\alpha })_{\alpha <\kappa }}"> </noscript><span class="lazy-image-placeholder" style="width: 7.933ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f7ba81b2efe577fa80b011abed68f41b7de827" data-alt="{\displaystyle (u_{\alpha })_{\alpha <\kappa }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa \leq \lambda }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> κ </mi> <mo> ≤ </mo> <mi> λ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \kappa \leq \lambda } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4905ac5ab29e59a511710130ca5712f1b910639" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.793ex; height:2.343ex;" alt="{\displaystyle \kappa \leq \lambda }"> </noscript><span class="lazy-image-placeholder" style="width: 5.793ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c4905ac5ab29e59a511710130ca5712f1b910639" data-alt="{\displaystyle \kappa \leq \lambda }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  such that for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \alpha \leq \lambda }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> α </mi> <mo> ≤ </mo> <mi> λ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \alpha \leq \lambda } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/815e1e30344b5cd134c8aeb1d0568880a436ab9b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:5.941ex; height:2.343ex;" alt="{\displaystyle \alpha \leq \lambda }"> </noscript><span class="lazy-image-placeholder" style="width: 5.941ex;height: 2.343ex;vertical-align: -0.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/815e1e30344b5cd134c8aeb1d0568880a436ab9b" data-alt="{\displaystyle \alpha \leq \lambda }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Complete_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Completion" class="mw-redirect" title="Complete space">completion</a> of the span of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{u_{\beta }:\beta <\min(\alpha ,\kappa )\}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> β </mi> </mrow> </msub> <mo> : </mo> <mi> β </mi> <mo> < </mo> <mo movablelimits="true" form="prefix"> min </mo> <mo stretchy="false"> ( </mo> <mi> α </mi> <mo> , </mo> <mi> κ </mi> <mo stretchy="false"> ) </mo> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{u_{\beta }:\beta <\min(\alpha ,\kappa )\}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206741efc3d54608f5926a065af41b9d67a3f8dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:20.741ex; height:3.009ex;" alt="{\displaystyle \{u_{\beta }:\beta <\min(\alpha ,\kappa )\}}"> </noscript><span class="lazy-image-placeholder" style="width: 20.741ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/206741efc3d54608f5926a065af41b9d67a3f8dc" data-alt="{\displaystyle \{u_{\beta }:\beta <\min(\alpha ,\kappa )\}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the same as that of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{v_{\beta }:\beta <\alpha \}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> { </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> β </mi> </mrow> </msub> <mo> : </mo> <mi> β </mi> <mo> < </mo> <mi> α </mi> <mo fence="false" stretchy="false"> } </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \{v_{\beta }:\beta <\alpha \}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85620dd5bcdfce16367ccc0338386b93642ea21c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.482ex; height:3.009ex;" alt="{\displaystyle \{v_{\beta }:\beta <\alpha \}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.482ex;height: 3.009ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/85620dd5bcdfce16367ccc0338386b93642ea21c" data-alt="{\displaystyle \{v_{\beta }:\beta <\alpha \}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In particular, when applied to a (algebraic) basis of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hilbert_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hilbert space">Hilbert space</a> (or, more generally, a basis of any dense subspace), it yields a (functional-analytic) orthonormal basis. Note that in the general case often the strict inequality <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \kappa <\lambda }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> κ </mi> <mo> < </mo> <mi> λ </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \kappa <\lambda } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97903fcd3076d6ae543501c10d80a37497892e6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.793ex; height:2.176ex;" alt="{\displaystyle \kappa <\lambda }"> </noscript><span class="lazy-image-placeholder" style="width: 5.793ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d97903fcd3076d6ae543501c10d80a37497892e6" data-alt="{\displaystyle \kappa <\lambda }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  holds, even if the starting set was linearly independent, and the span of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (u_{\alpha })_{\alpha <\kappa }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> u </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> </mrow> </msub> <msub> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> <mo> < </mo> <mi> κ </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (u_{\alpha })_{\alpha <\kappa }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f7ba81b2efe577fa80b011abed68f41b7de827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.933ex; height:2.843ex;" alt="{\displaystyle (u_{\alpha })_{\alpha <\kappa }}"> </noscript><span class="lazy-image-placeholder" style="width: 7.933ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f7ba81b2efe577fa80b011abed68f41b7de827" data-alt="{\displaystyle (u_{\alpha })_{\alpha <\kappa }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  need not be a subspace of the span of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{\alpha })_{\alpha <\lambda }}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false"> ( </mo> <msub> <mi> v </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> </mrow> </msub> <msub> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> α </mi> <mo> < </mo> <mi> λ </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle (v_{\alpha })_{\alpha <\lambda }} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb44bc2105d877ca44d2a4c75e35fe5b71acceb" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.742ex; height:2.843ex;" alt="{\displaystyle (v_{\alpha })_{\alpha <\lambda }}"> </noscript><span class="lazy-image-placeholder" style="width: 7.742ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6bb44bc2105d877ca44d2a4c75e35fe5b71acceb" data-alt="{\displaystyle (v_{\alpha })_{\alpha <\lambda }}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (rather, it's a subspace of its completion). </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(2)"> <h2 id="Example">Example</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=2&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Example" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-2 collapsible-block" id="mf-section-2"> <div class="mw-heading mw-heading3"> <h3 id="Euclidean_space">Euclidean space</h3> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=3&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Euclidean space" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> Consider the following set of vectors in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {R} ^{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbb {R} ^{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.732ex; height:2.676ex;" alt="{\displaystyle \mathbb {R} ^{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.732ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e150115ab9f63023215109595b76686a1ff890fd" data-alt="{\displaystyle \mathbb {R} ^{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  (with the conventional <a href="https://en-m-wikipedia-org.translate.goog/wiki/Inner_product_space?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Euclidean_vector_space" title="Inner product space">inner product</a>) <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S=\left\{\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2\\2\end{bmatrix}}\right\}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> S </mi> <mo> = </mo> <mrow> <mo> { </mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </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> 3 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 2 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 2 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mrow> <mo> } </mo> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle S=\left\{\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2\\2\end{bmatrix}}\right\}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2603636bc33e0dc49d512a31d93cee6df06251b9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.017ex; height:6.176ex;" alt="{\displaystyle S=\left\{\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2\\2\end{bmatrix}}\right\}.}"> </noscript><span class="lazy-image-placeholder" style="width: 30.017ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2603636bc33e0dc49d512a31d93cee6df06251b9" data-alt="{\displaystyle S=\left\{\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2\\2\end{bmatrix}}\right\}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Now, perform Gram–Schmidt, to obtain an orthogonal set of vectors: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1}={\begin{bmatrix}3\\1\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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </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> 3 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ea071c47dc8e591ea9e271ac27e1eac8ed2e0d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:15.571ex; height:6.176ex;" alt="{\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 15.571ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1ea071c47dc8e591ea9e271ac27e1eac8ed2e0d" data-alt="{\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1}={\begin{bmatrix}3\\1\end{bmatrix}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2})={\begin{bmatrix}2\\2\end{bmatrix}}-\operatorname {proj} _{\left[{\begin{smallmatrix}3\\1\end{smallmatrix}}\right]}{\begin{bmatrix}2\\2\end{bmatrix}}={\begin{bmatrix}2\\2\end{bmatrix}}-{\frac {8}{10}}{\begin{bmatrix}3\\1\end{bmatrix}}={\begin{bmatrix}-2/5\\6/5\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> 2 </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> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 2 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 2 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle scriptlevel="1"> <mtable rowspacing=".2em" columnspacing="0.333em" displaystyle="false"> <mtr> <mtd> <mn> 3 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> </mtr> </mtable> </mstyle> </mrow> <mo> ] </mo> </mrow> </mrow> </msub> <mo> ⁡ </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 2 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 2 </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> 2 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 2 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 8 </mn> <mn> 10 </mn> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 3 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 1 </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> <mo> − </mo> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 6 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2})={\begin{bmatrix}2\\2\end{bmatrix}}-\operatorname {proj} _{\left[{\begin{smallmatrix}3\\1\end{smallmatrix}}\right]}{\begin{bmatrix}2\\2\end{bmatrix}}={\begin{bmatrix}2\\2\end{bmatrix}}-{\frac {8}{10}}{\begin{bmatrix}3\\1\end{bmatrix}}={\begin{bmatrix}-2/5\\6/5\end{bmatrix}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d583d79b8ab535ab1b2bd0c5bd7960f02f743f41" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:73.996ex; height:6.176ex;" alt="{\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2})={\begin{bmatrix}2\\2\end{bmatrix}}-\operatorname {proj} _{\left[{\begin{smallmatrix}3\\1\end{smallmatrix}}\right]}{\begin{bmatrix}2\\2\end{bmatrix}}={\begin{bmatrix}2\\2\end{bmatrix}}-{\frac {8}{10}}{\begin{bmatrix}3\\1\end{bmatrix}}={\begin{bmatrix}-2/5\\6/5\end{bmatrix}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 73.996ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d583d79b8ab535ab1b2bd0c5bd7960f02f743f41" data-alt="{\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{2})={\begin{bmatrix}2\\2\end{bmatrix}}-\operatorname {proj} _{\left[{\begin{smallmatrix}3\\1\end{smallmatrix}}\right]}{\begin{bmatrix}2\\2\end{bmatrix}}={\begin{bmatrix}2\\2\end{bmatrix}}-{\frac {8}{10}}{\begin{bmatrix}3\\1\end{bmatrix}}={\begin{bmatrix}-2/5\\6/5\end{bmatrix}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  We check that the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1}}"><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> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb86457c0ac456302419c967bda92a5910ff004c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.54ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.54ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb86457c0ac456302419c967bda92a5910ff004c" data-alt="{\displaystyle \mathbf {u} _{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{2}}"><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> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636f3f24b87f95f1c772c4ac0b579c9ee7b43899" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.54ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.54ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/636f3f24b87f95f1c772c4ac0b579c9ee7b43899" data-alt="{\displaystyle \mathbf {u} _{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are indeed orthogonal: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle =\left\langle {\begin{bmatrix}3\\1\end{bmatrix}},{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}\right\rangle =-{\frac {6}{5}}+{\frac {6}{5}}=0,}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> <mo> = </mo> <mrow> <mo> ⟨ </mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 3 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 1 </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> <mo> − </mo> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 6 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mrow> <mo> ⟩ </mo> </mrow> <mo> = </mo> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 6 </mn> <mn> 5 </mn> </mfrac> </mrow> <mo> + </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 6 </mn> <mn> 5 </mn> </mfrac> </mrow> <mo> = </mo> <mn> 0 </mn> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle =\left\langle {\begin{bmatrix}3\\1\end{bmatrix}},{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}\right\rangle =-{\frac {6}{5}}+{\frac {6}{5}}=0,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4583cbf73f2fc900aaf8917009315a1f9617ff" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:45.064ex; height:6.176ex;" alt="{\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle =\left\langle {\begin{bmatrix}3\\1\end{bmatrix}},{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}\right\rangle =-{\frac {6}{5}}+{\frac {6}{5}}=0,}"> </noscript><span class="lazy-image-placeholder" style="width: 45.064ex;height: 6.176ex;vertical-align: -2.505ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fb4583cbf73f2fc900aaf8917009315a1f9617ff" data-alt="{\displaystyle \langle \mathbf {u} _{1},\mathbf {u} _{2}\rangle =\left\langle {\begin{bmatrix}3\\1\end{bmatrix}},{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}\right\rangle =-{\frac {6}{5}}+{\frac {6}{5}}=0,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  noting that if the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Dot_product?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Dot product">dot product</a> of two vectors is 0 then they are orthogonal. For non-zero vectors, we can then normalize the vectors by dividing out their sizes as shown above: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3\\1\end{bmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <mn> 10 </mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 3 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3\\1\end{bmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b378b40503fe198b4591f55b0e3ae32ed2b993" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:14.844ex; height:6.509ex;" alt="{\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3\\1\end{bmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 14.844ex;height: 6.509ex;vertical-align: -2.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c6b378b40503fe198b4591f55b0e3ae32ed2b993" data-alt="{\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3\\1\end{bmatrix}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {40 \over 25}}}{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1\\3\end{bmatrix}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <mfrac> <mn> 40 </mn> <mn> 25 </mn> </mfrac> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <mn> 2 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 6 </mn> <mrow class="MJX-TeXAtom-ORD"> <mo> / </mo> </mrow> <mn> 5 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <mn> 10 </mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <mn> 1 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 3 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {40 \over 25}}}{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1\\3\end{bmatrix}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43be2370ce8ac1fa33165e613116ba269317ac4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.671ex; width:34.539ex; height:8.343ex;" alt="{\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {40 \over 25}}}{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1\\3\end{bmatrix}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 34.539ex;height: 8.343ex;vertical-align: -4.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/43be2370ce8ac1fa33165e613116ba269317ac4a" data-alt="{\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {40 \over 25}}}{\begin{bmatrix}-2/5\\6/5\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1\\3\end{bmatrix}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(3)"> <h2 id="Properties">Properties</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=4&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Properties" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-3 collapsible-block" id="mf-section-3"> Denote by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> GS </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32352a2d0e05a4a22f8796672311bc79b67c09ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.069ex; height:2.843ex;" alt="{\displaystyle \operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})}"> </noscript><span class="lazy-image-placeholder" style="width: 15.069ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32352a2d0e05a4a22f8796672311bc79b67c09ff" data-alt="{\displaystyle \operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  the result of applying the Gram–Schmidt process to a collection of vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94fad4acc15e19543d874b8141475083d18229f4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.143ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.143ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/94fad4acc15e19543d874b8141475083d18229f4" data-alt="{\displaystyle \mathbf {v} _{1},\dots ,\mathbf {v} _{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This yields a map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GS} \colon (\mathbb {R} ^{n})^{k}\to (\mathbb {R} ^{n})^{k}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> GS </mi> <mo> : </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> <mo stretchy="false"> → </mo> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {GS} \colon (\mathbb {R} ^{n})^{k}\to (\mathbb {R} ^{n})^{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f551da3793727af52ef93dba678068dbab55e79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.354ex; height:3.176ex;" alt="{\displaystyle \operatorname {GS} \colon (\mathbb {R} ^{n})^{k}\to (\mathbb {R} ^{n})^{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 19.354ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f551da3793727af52ef93dba678068dbab55e79" data-alt="{\displaystyle \operatorname {GS} \colon (\mathbb {R} ^{n})^{k}\to (\mathbb {R} ^{n})^{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . It has the following properties: <ul> <li>It is continuous</li> <li>It is <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orientation_(vector_space)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orientation (vector space)">orientation</a> preserving in the sense that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {or} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})=\operatorname {or} (\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k}))}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> or </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> = </mo> <mi> or </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> GS </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {or} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})=\operatorname {or} (\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k}))} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c02d92df291aaa74c25e02f6d225ab62cf860c64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:36.078ex; height:2.843ex;" alt="{\displaystyle \operatorname {or} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})=\operatorname {or} (\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k}))}"> </noscript><span class="lazy-image-placeholder" style="width: 36.078ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c02d92df291aaa74c25e02f6d225ab62cf860c64" data-alt="{\displaystyle \operatorname {or} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})=\operatorname {or} (\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k}))}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> .</li> <li>It commutes with orthogonal maps:</li> </ul> Let <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> g </mi> <mo> : </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo stretchy="false"> → </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f147e7d7a52fb587d2eb46a4a767351ecfe6c5f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.557ex; height:2.676ex;" alt="{\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 11.557ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f147e7d7a52fb587d2eb46a4a767351ecfe6c5f" data-alt="{\displaystyle g\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  be orthogonal (with respect to the given inner product). Then we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GS} (g(\mathbf {v} _{1}),\dots ,g(\mathbf {v} _{k}))=\left(g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{1}),\dots ,g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{k})\right)}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> GS </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow> <mo> ( </mo> <mrow> <mi> g </mi> <mo stretchy="false"> ( </mo> <mi> GS </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> <mo> … </mo> <mo> , </mo> <mi> g </mi> <mo stretchy="false"> ( </mo> <mi> GS </mi> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <msub> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> </mrow> <mo> ) </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {GS} (g(\mathbf {v} _{1}),\dots ,g(\mathbf {v} _{k}))=\left(g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{1}),\dots ,g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{k})\right)} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5d13effe8b8311422ae188835c3d44fdb2c3ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:69.138ex; height:2.843ex;" alt="{\displaystyle \operatorname {GS} (g(\mathbf {v} _{1}),\dots ,g(\mathbf {v} _{k}))=\left(g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{1}),\dots ,g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{k})\right)}"> </noscript><span class="lazy-image-placeholder" style="width: 69.138ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b5d13effe8b8311422ae188835c3d44fdb2c3ac" data-alt="{\displaystyle \operatorname {GS} (g(\mathbf {v} _{1}),\dots ,g(\mathbf {v} _{k}))=\left(g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{1}),\dots ,g(\operatorname {GS} (\mathbf {v} _{1},\dots ,\mathbf {v} _{k})_{k})\right)}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Further, a parametrized version of the Gram–Schmidt process yields a (strong) <a href="https://en-m-wikipedia-org.translate.goog/wiki/Retraction_(topology)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#Deformation_retract_and_strong_deformation_retract" title="Retraction (topology)">deformation retraction</a> of the general linear group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {GL} (\mathbb {R} ^{n})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal"> G </mi> <mi mathvariant="normal"> L </mi> </mrow> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathrm {GL} (\mathbb {R} ^{n})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59bb5243e62a46a816dac70e7e5f737c5853a130" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.983ex; height:2.843ex;" alt="{\displaystyle \mathrm {GL} (\mathbb {R} ^{n})}"> </noscript><span class="lazy-image-placeholder" style="width: 7.983ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/59bb5243e62a46a816dac70e7e5f737c5853a130" data-alt="{\displaystyle \mathrm {GL} (\mathbb {R} ^{n})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  onto the orthogonal group <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O(\mathbb {R} ^{n})}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> O </mi> <mo stretchy="false"> ( </mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck"> R </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msup> <mo stretchy="false"> ) </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle O(\mathbb {R} ^{n})} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8cb4eeb172b54cd039a318a72e4f5701b2b376" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.479ex; height:2.843ex;" alt="{\displaystyle O(\mathbb {R} ^{n})}"> </noscript><span class="lazy-image-placeholder" style="width: 6.479ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f8cb4eeb172b54cd039a318a72e4f5701b2b376" data-alt="{\displaystyle O(\mathbb {R} ^{n})}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(4)"> <h2 id="Numerical_stability">Numerical stability</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=5&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Numerical stability" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-4 collapsible-block" id="mf-section-4"> When this process is implemented on a computer, the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{k}}"><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"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6db1dab97ca1af730a50c30421af8c3fd020c836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.574ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6db1dab97ca1af730a50c30421af8c3fd020c836" data-alt="{\displaystyle \mathbf {u} _{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are often not quite orthogonal, due to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Round-off_error?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Round-off error">rounding errors</a>. For the Gram–Schmidt process as described above (sometimes referred to as "classical Gram–Schmidt") this loss of orthogonality is particularly bad; therefore, it is said that the (classical) Gram–Schmidt process is <a href="https://en-m-wikipedia-org.translate.goog/wiki/Numerical_stability?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Numerical stability">numerically unstable</a>. The Gram–Schmidt process can be stabilized by a small modification; this version is sometimes referred to as modified Gram-Schmidt or MGS. This approach gives the same result as the original formula in exact arithmetic and introduces smaller errors in finite-precision arithmetic. Instead of computing the vector uk as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{k})-\cdots -\operatorname {proj} _{\mathbf {u} _{k-1}}(\mathbf {v} _{k}),}"><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"> <mi> k </mi> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> − </mo> <mo> ⋯ </mo> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{k})-\cdots -\operatorname {proj} _{\mathbf {u} _{k-1}}(\mathbf {v} _{k}),} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd103ec51b97bfcb55150768b553f57367af588" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:56.142ex; height:3.343ex;" alt="{\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{k})-\cdots -\operatorname {proj} _{\mathbf {u} _{k-1}}(\mathbf {v} _{k}),}"> </noscript><span class="lazy-image-placeholder" style="width: 56.142ex;height: 3.343ex;vertical-align: -1.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6cd103ec51b97bfcb55150768b553f57367af588" data-alt="{\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k})-\operatorname {proj} _{\mathbf {u} _{2}}(\mathbf {v} _{k})-\cdots -\operatorname {proj} _{\mathbf {u} _{k-1}}(\mathbf {v} _{k}),}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  it is computed as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\mathbf {u} _{k}^{(1)}&=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k}),\\\mathbf {u} _{k}^{(2)}&=\mathbf {u} _{k}^{(1)}-\operatorname {proj} _{\mathbf {u} _{2}}\left(\mathbf {u} _{k}^{(1)}\right),\\&\;\;\vdots \\\mathbf {u} _{k}^{(k-2)}&=\mathbf {u} _{k}^{(k-3)}-\operatorname {proj} _{\mathbf {u} _{k-2}}\left(\mathbf {u} _{k}^{(k-3)}\right),\\\mathbf {u} _{k}^{(k-1)}&=\mathbf {u} _{k}^{(k-2)}-\operatorname {proj} _{\mathbf {u} _{k-1}}\left(\mathbf {u} _{k}^{(k-2)}\right),\\\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}^{(k-1)}}{\left\|\mathbf {u} _{k}^{(k-1)}\right\|}}\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> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo> = </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> ) </mo> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd></mtd> <mtd> <mi></mi> <mspace width="thickmathspace"></mspace> <mspace width="thickmathspace"></mspace> <mo> ⋮ </mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo> = </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> − </mo> <mn> 2 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 3 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> ) </mo> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mtd> <mtd> <mi></mi> <mo> = </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> − </mo> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> </mrow> </msub> <mo> ⁡ </mo> <mrow> <mo> ( </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> ) </mo> </mrow> <mo> , </mo> </mtd> </mtr> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> </mtd> <mtd> <mi></mi> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mrow> <mo symmetric="true"> ‖ </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo symmetric="true"> ‖ </mo> </mrow> </mfrac> </mrow> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle {\begin{aligned}\mathbf {u} _{k}^{(1)}&=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k}),\\\mathbf {u} _{k}^{(2)}&=\mathbf {u} _{k}^{(1)}-\operatorname {proj} _{\mathbf {u} _{2}}\left(\mathbf {u} _{k}^{(1)}\right),\\&\;\;\vdots \\\mathbf {u} _{k}^{(k-2)}&=\mathbf {u} _{k}^{(k-3)}-\operatorname {proj} _{\mathbf {u} _{k-2}}\left(\mathbf {u} _{k}^{(k-3)}\right),\\\mathbf {u} _{k}^{(k-1)}&=\mathbf {u} _{k}^{(k-2)}-\operatorname {proj} _{\mathbf {u} _{k-1}}\left(\mathbf {u} _{k}^{(k-2)}\right),\\\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}^{(k-1)}}{\left\|\mathbf {u} _{k}^{(k-1)}\right\|}}\end{aligned}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e130edad422fde0487a416b42a3dde83693fe1c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -14.947ex; margin-bottom: -0.224ex; width:35.9ex; height:31.509ex;" alt="{\displaystyle {\begin{aligned}\mathbf {u} _{k}^{(1)}&=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k}),\\\mathbf {u} _{k}^{(2)}&=\mathbf {u} _{k}^{(1)}-\operatorname {proj} _{\mathbf {u} _{2}}\left(\mathbf {u} _{k}^{(1)}\right),\\&\;\;\vdots \\\mathbf {u} _{k}^{(k-2)}&=\mathbf {u} _{k}^{(k-3)}-\operatorname {proj} _{\mathbf {u} _{k-2}}\left(\mathbf {u} _{k}^{(k-3)}\right),\\\mathbf {u} _{k}^{(k-1)}&=\mathbf {u} _{k}^{(k-2)}-\operatorname {proj} _{\mathbf {u} _{k-1}}\left(\mathbf {u} _{k}^{(k-2)}\right),\\\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}^{(k-1)}}{\left\|\mathbf {u} _{k}^{(k-1)}\right\|}}\end{aligned}}}"> </noscript><span class="lazy-image-placeholder" style="width: 35.9ex;height: 31.509ex;vertical-align: -14.947ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e130edad422fde0487a416b42a3dde83693fe1c" data-alt="{\displaystyle {\begin{aligned}\mathbf {u} _{k}^{(1)}&=\mathbf {v} _{k}-\operatorname {proj} _{\mathbf {u} _{1}}(\mathbf {v} _{k}),\\\mathbf {u} _{k}^{(2)}&=\mathbf {u} _{k}^{(1)}-\operatorname {proj} _{\mathbf {u} _{2}}\left(\mathbf {u} _{k}^{(1)}\right),\\&\;\;\vdots \\\mathbf {u} _{k}^{(k-2)}&=\mathbf {u} _{k}^{(k-3)}-\operatorname {proj} _{\mathbf {u} _{k-2}}\left(\mathbf {u} _{k}^{(k-3)}\right),\\\mathbf {u} _{k}^{(k-1)}&=\mathbf {u} _{k}^{(k-2)}-\operatorname {proj} _{\mathbf {u} _{k-1}}\left(\mathbf {u} _{k}^{(k-2)}\right),\\\mathbf {e} _{k}&={\frac {\mathbf {u} _{k}^{(k-1)}}{\left\|\mathbf {u} _{k}^{(k-1)}\right\|}}\end{aligned}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  This method is used in the previous animation, when the intermediate <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"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> <mo> ′ </mo> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} '_{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0942ad4902a2632037eb59a2d3650af22db2115b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.465ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} '_{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.465ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0942ad4902a2632037eb59a2d3650af22db2115b" data-alt="{\displaystyle \mathbf {v} '_{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  vector is used when orthogonalizing the blue vector <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> <noscript> <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}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.465ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edbc3fb88d6a27517dc79da85446ab8147801be9" data-alt="{\displaystyle \mathbf {v} _{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Here is another description of the modified algorithm. Given the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921a06d96c2e9632ae9bb14077091d46c9f7c58f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.772ex; height:2.009ex;" alt="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.772ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/921a06d96c2e9632ae9bb14077091d46c9f7c58f" data-alt="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2},\dots ,\mathbf {v} _{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , in our first step we produce vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\dots ,\mathbf {v} _{n}^{(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> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c3cae30e175411a7f00653eec218eac5a7f00c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.167ex; height:3.676ex;" alt="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}}"> </noscript><span class="lazy-image-placeholder" style="width: 16.167ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/81c3cae30e175411a7f00653eec218eac5a7f00c" data-alt="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> by removing components along the direction of <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> <noscript> <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}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.465ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/282458bb19c231f94697405bddd93af04a34cabe" data-alt="{\displaystyle \mathbf {v} _{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . In formulas, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{k}^{(1)}:=\mathbf {v} _{k}-{\frac {\langle \mathbf {v} _{k},\mathbf {v} _{1}\rangle }{\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle }}\mathbf {v} _{1}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> := </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mrow> <mrow> <mo fence="false" stretchy="false"> ⟨ </mo> <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> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mrow> </mfrac> </mrow> <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} _{k}^{(1)}:=\mathbf {v} _{k}-{\frac {\langle \mathbf {v} _{k},\mathbf {v} _{1}\rangle }{\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle }}\mathbf {v} _{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e099e4386cae481efe912ed95a30200347309b3c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:23.939ex; height:6.509ex;" alt="{\displaystyle \mathbf {v} _{k}^{(1)}:=\mathbf {v} _{k}-{\frac {\langle \mathbf {v} _{k},\mathbf {v} _{1}\rangle }{\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle }}\mathbf {v} _{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 23.939ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e099e4386cae481efe912ed95a30200347309b3c" data-alt="{\displaystyle \mathbf {v} _{k}^{(1)}:=\mathbf {v} _{k}-{\frac {\langle \mathbf {v} _{k},\mathbf {v} _{1}\rangle }{\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle }}\mathbf {v} _{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . After this step we already have two of our desired orthogonal vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25058c5d748ce5cef7fa04c15166242860d6a3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.422ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.422ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25058c5d748ce5cef7fa04c15166242860d6a3fa" data-alt="{\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , namely <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1},\mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}}"><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> <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"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1},\mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c062225d1a1594bc3732f6318685c0409f720115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.52ex; height:3.676ex;" alt="{\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1},\mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}}"> </noscript><span class="lazy-image-placeholder" style="width: 18.52ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c062225d1a1594bc3732f6318685c0409f720115" data-alt="{\displaystyle \mathbf {u} _{1}=\mathbf {v} _{1},\mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , but we also made <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{3}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{3}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1fceb05a230101b36a4a1b3e936524afafb4fea" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.667ex; height:3.676ex;" alt="{\displaystyle \mathbf {v} _{3}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.667ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b1fceb05a230101b36a4a1b3e936524afafb4fea" data-alt="{\displaystyle \mathbf {v} _{3}^{(1)},\dots ,\mathbf {v} _{n}^{(1)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  already orthogonal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1}}"><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> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb86457c0ac456302419c967bda92a5910ff004c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.54ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.54ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb86457c0ac456302419c967bda92a5910ff004c" data-alt="{\displaystyle \mathbf {u} _{1}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Next, we orthogonalize those remaining vectors against <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}}"><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> 2 </mn> </mrow> </msub> <mo> = </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51d3f41916742ed0477de7ffe2d116b951efaec8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.383ex; height:3.676ex;" alt="{\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.383ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51d3f41916742ed0477de7ffe2d116b951efaec8" data-alt="{\displaystyle \mathbf {u} _{2}=\mathbf {v} _{2}^{(1)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . This means we compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd076cc965383a35deb8f265bb01182d72eb829" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.446ex; height:3.676ex;" alt="{\displaystyle \mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}}"> </noscript><span class="lazy-image-placeholder" style="width: 17.446ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bfd076cc965383a35deb8f265bb01182d72eb829" data-alt="{\displaystyle \mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  by subtraction <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{k}^{(2)}:=\mathbf {v} _{k}^{(1)}-{\frac {\langle \mathbf {v} _{k}^{(1)},\mathbf {u} _{2}\rangle }{\langle \mathbf {u} _{2},\mathbf {u} _{2}\rangle }}\mathbf {u} _{2}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> := </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> − </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" stretchy="false"> ⟨ </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mrow> <mrow> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mrow> </mfrac> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{k}^{(2)}:=\mathbf {v} _{k}^{(1)}-{\frac {\langle \mathbf {v} _{k}^{(1)},\mathbf {u} _{2}\rangle }{\langle \mathbf {u} _{2},\mathbf {u} _{2}\rangle }}\mathbf {u} _{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79410284a17e221a87bbb4a3e6cde790339bdd18" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:26.578ex; height:7.343ex;" alt="{\displaystyle \mathbf {v} _{k}^{(2)}:=\mathbf {v} _{k}^{(1)}-{\frac {\langle \mathbf {v} _{k}^{(1)},\mathbf {u} _{2}\rangle }{\langle \mathbf {u} _{2},\mathbf {u} _{2}\rangle }}\mathbf {u} _{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 26.578ex;height: 7.343ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/79410284a17e221a87bbb4a3e6cde790339bdd18" data-alt="{\displaystyle \mathbf {v} _{k}^{(2)}:=\mathbf {v} _{k}^{(1)}-{\frac {\langle \mathbf {v} _{k}^{(1)},\mathbf {u} _{2}\rangle }{\langle \mathbf {u} _{2},\mathbf {u} _{2}\rangle }}\mathbf {u} _{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Now we have stored the vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(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> 1 </mn> </mrow> </msub> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 1 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ac750c595d32baee48cf30701ffb8a2f379a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:25.724ex; height:3.676ex;" alt="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}}"> </noscript><span class="lazy-image-placeholder" style="width: 25.724ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a1ac750c595d32baee48cf30701ffb8a2f379a1e" data-alt="{\displaystyle \mathbf {v} _{1},\mathbf {v} _{2}^{(1)},\mathbf {v} _{3}^{(2)},\mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  where the first three vectors are already <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\mathbf {u} _{3}}"><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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\mathbf {u} _{3}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46acf47fb7bf3b2a9e4ab3b386112907373173ad" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.686ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\mathbf {u} _{3}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.686ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46acf47fb7bf3b2a9e4ab3b386112907373173ad" data-alt="{\displaystyle \mathbf {u} _{1},\mathbf {u} _{2},\mathbf {u} _{3}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and the remaining vectors are already orthogonal to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1},\mathbf {u} _{2}}"><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> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1},\mathbf {u} _{2}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3248d4f38ea20f4861745767390f05838b6fcfa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.113ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\mathbf {u} _{2}}"> </noscript><span class="lazy-image-placeholder" style="width: 6.113ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3248d4f38ea20f4861745767390f05838b6fcfa" data-alt="{\displaystyle \mathbf {u} _{1},\mathbf {u} _{2}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . As should be clear now, the next step orthogonalizes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 4 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> <mo> , </mo> <mo> … </mo> <mo> , </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20b9e4f17b9f0b777e928a2c1688b17623cbbbe" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:12.667ex; height:3.676ex;" alt="{\displaystyle \mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}}"> </noscript><span class="lazy-image-placeholder" style="width: 12.667ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e20b9e4f17b9f0b777e928a2c1688b17623cbbbe" data-alt="{\displaystyle \mathbf {v} _{4}^{(2)},\dots ,\mathbf {v} _{n}^{(2)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  against <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{3}=\mathbf {v} _{3}^{(2)}}"><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> 3 </mn> </mrow> </msub> <mo> = </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 3 </mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> ( </mo> <mn> 2 </mn> <mo stretchy="false"> ) </mo> </mrow> </msubsup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{3}=\mathbf {v} _{3}^{(2)}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e351e4f3a9064d4f98641f5911ec336184198c87" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.383ex; height:3.676ex;" alt="{\displaystyle \mathbf {u} _{3}=\mathbf {v} _{3}^{(2)}}"> </noscript><span class="lazy-image-placeholder" style="width: 9.383ex;height: 3.676ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e351e4f3a9064d4f98641f5911ec336184198c87" data-alt="{\displaystyle \mathbf {u} _{3}=\mathbf {v} _{3}^{(2)}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . Proceeding in this manner we find the full set of orthogonal vectors <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}"><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> <mo> … </mo> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> n </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25058c5d748ce5cef7fa04c15166242860d6a3fa" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.422ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}"> </noscript><span class="lazy-image-placeholder" style="width: 10.422ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25058c5d748ce5cef7fa04c15166242860d6a3fa" data-alt="{\displaystyle \mathbf {u} _{1},\dots ,\mathbf {u} _{n}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . If orthonormal vectors are desired, then we normalize as we go, so that the denominators in the subtraction formulas turn into ones. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(5)"> <h2 id="Algorithm">Algorithm</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=6&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Algorithm" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-5 collapsible-block" id="mf-section-5"> The following <a href="https://en-m-wikipedia-org.translate.goog/wiki/MATLAB?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="MATLAB">MATLAB</a> algorithm implements classical Gram–Schmidt orthonormalization. The vectors v1, ..., vk (columns of matrix <code>V</code>, so that <code>V(:,j)</code> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> j </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle j} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"> </noscript><span class="lazy-image-placeholder" style="width: 0.985ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" data-alt="{\displaystyle j}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> th vector) are replaced by orthonormal vectors (columns of <code>U</code>) which span the same subspace. <div class="mw-highlight mw-highlight-lang-matlab mw-content-ltr mw-highlight-lines" dir="ltr"> <pre>function U = gramschmidt(V) [n, k] = size(V); U = zeros(n,k); U(:,1) = V(:,1) / norm(V(:,1)); for i = 2:k U(:,i) = V(:,i); for j = 1:i-1 U(:,i) = U(:,i) - (U(:,j)'*U(:,i)) * U(:,j); end U(:,i) = U(:,i) / norm(U(:,i)); end end </pre> </div> The cost of this algorithm is asymptotically O(nk2) floating point operations, where n is the dimensionality of the vectors.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-FOOTNOTEGolubVan_Loan1996%C2%A75.2.8-3">[2]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(6)"> <h2 id="Via_Gaussian_elimination">Via Gaussian elimination</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=7&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Via Gaussian elimination" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-6 collapsible-block" id="mf-section-6"> If the rows {v1, ..., vk} are written as 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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , then applying <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gaussian_elimination?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gaussian elimination">Gaussian elimination</a> to the augmented matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[AA^{\mathsf {T}}|A\right]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> [ </mo> <mrow> <mi> A </mi> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> A </mi> </mrow> <mo> ] </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left[AA^{\mathsf {T}}|A\right]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01fe20a09baca8ac8933fb783fdc7908d083190e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.167ex; height:3.343ex;" alt="{\displaystyle \left[AA^{\mathsf {T}}|A\right]}"> </noscript><span class="lazy-image-placeholder" style="width: 9.167ex;height: 3.343ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/01fe20a09baca8ac8933fb783fdc7908d083190e" data-alt="{\displaystyle \left[AA^{\mathsf {T}}|A\right]}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  will produce the orthogonalized vectors in place 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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.743ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" data-alt="{\displaystyle A}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . However the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle AA^{\mathsf {T}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> A </mi> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle AA^{\mathsf {T}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af621dc24b8fc32577441725e8a8a112508ebcd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.838ex; height:2.676ex;" alt="{\displaystyle AA^{\mathsf {T}}}"> </noscript><span class="lazy-image-placeholder" style="width: 4.838ex;height: 2.676ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af621dc24b8fc32577441725e8a8a112508ebcd6" data-alt="{\displaystyle AA^{\mathsf {T}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  must be brought to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Row_echelon_form?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Row echelon form">row echelon form</a>, using only the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Elementary_matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Elementary matrix">row operation</a> of adding a scalar multiple of one row to another.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-4">[3]</a> For example, taking <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {v} _{1}={\begin{bmatrix}3&1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2&2\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"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 3 </mn> </mtd> <mtd> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <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> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 2 </mn> </mtd> <mtd> <mn> 2 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {v} _{1}={\begin{bmatrix}3&1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2&2\end{bmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768ad151c5b923ceb2d388004d5de026d0a0e2f5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.547ex; height:2.843ex;" alt="{\displaystyle \mathbf {v} _{1}={\begin{bmatrix}3&1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2&2\end{bmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 25.547ex;height: 2.843ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/768ad151c5b923ceb2d388004d5de026d0a0e2f5" data-alt="{\displaystyle \mathbf {v} _{1}={\begin{bmatrix}3&1\end{bmatrix}},\mathbf {v} _{2}={\begin{bmatrix}2&2\end{bmatrix}}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  as above, we have <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[AA^{\mathsf {T}}|A\right]=\left[{\begin{array}{rr|rr}10&8&3&1\\8&8&2&2\end{array}}\right]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> [ </mo> <mrow> <mi> A </mi> <msup> <mi> A </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="sans-serif"> T </mi> </mrow> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false"> | </mo> </mrow> <mi> A </mi> </mrow> <mo> ] </mo> </mrow> <mo> = </mo> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none solid none"> <mtr> <mtd> <mn> 10 </mn> </mtd> <mtd> <mn> 8 </mn> </mtd> <mtd> <mn> 3 </mn> </mtd> <mtd> <mn> 1 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 8 </mn> </mtd> <mtd> <mn> 8 </mn> </mtd> <mtd> <mn> 2 </mn> </mtd> <mtd> <mn> 2 </mn> </mtd> </mtr> </mtable> </mrow> <mo> ] </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left[AA^{\mathsf {T}}|A\right]=\left[{\begin{array}{rr|rr}10&8&3&1\\8&8&2&2\end{array}}\right]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25852adcb609d5e10e2c3514816c83d562db6cfe" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:30.366ex; height:7.509ex;" alt="{\displaystyle \left[AA^{\mathsf {T}}|A\right]=\left[{\begin{array}{rr|rr}10&8&3&1\\8&8&2&2\end{array}}\right]}"> </noscript><span class="lazy-image-placeholder" style="width: 30.366ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/25852adcb609d5e10e2c3514816c83d562db6cfe" data-alt="{\displaystyle \left[AA^{\mathsf {T}}|A\right]=\left[{\begin{array}{rr|rr}10&8&3&1\\8&8&2&2\end{array}}\right]}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  And reducing this to <a href="https://en-m-wikipedia-org.translate.goog/wiki/Row_echelon_form?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Row echelon form">row echelon form</a> produces <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left[{\begin{array}{rr|rr}1&.8&.3&.1\\0&1&-.25&.75\end{array}}\right]}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo> [ </mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right" rowspacing="4pt" columnspacing="1em" columnlines="none solid none"> <mtr> <mtd> <mn> 1 </mn> </mtd> <mtd> <mn> .8 </mn> </mtd> <mtd> <mn> .3 </mn> </mtd> <mtd> <mn> .1 </mn> </mtd> </mtr> <mtr> <mtd> <mn> 0 </mn> </mtd> <mtd> <mn> 1 </mn> </mtd> <mtd> <mo> − </mo> <mn> .25 </mn> </mtd> <mtd> <mn> .75 </mn> </mtd> </mtr> </mtable> </mrow> <mo> ] </mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \left[{\begin{array}{rr|rr}1&.8&.3&.1\\0&1&-.25&.75\end{array}}\right]} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713fd6ba7de58764d341f799d9f2c91a3c1ac750" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:23.011ex; height:7.509ex;" alt="{\displaystyle \left[{\begin{array}{rr|rr}1&.8&.3&.1\\0&1&-.25&.75\end{array}}\right]}"> </noscript><span class="lazy-image-placeholder" style="width: 23.011ex;height: 7.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/713fd6ba7de58764d341f799d9f2c91a3c1ac750" data-alt="{\displaystyle \left[{\begin{array}{rr|rr}1&.8&.3&.1\\0&1&-.25&.75\end{array}}\right]}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  The normalized vectors are then <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {.3^{2}+.1^{2}}}}{\begin{bmatrix}.3&.1\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3&1\end{bmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <msup> <mn> .3 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mn> .1 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> .3 </mn> </mtd> <mtd> <mn> .1 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <mn> 10 </mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn> 3 </mn> </mtd> <mtd> <mn> 1 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {.3^{2}+.1^{2}}}}{\begin{bmatrix}.3&.1\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3&1\end{bmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30019eb8f368f6ac0e76902fcf9c8deaabf6b8a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:39.98ex; height:6.509ex;" alt="{\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {.3^{2}+.1^{2}}}}{\begin{bmatrix}.3&.1\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3&1\end{bmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 39.98ex;height: 6.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/30019eb8f368f6ac0e76902fcf9c8deaabf6b8a8" data-alt="{\displaystyle \mathbf {e} _{1}={\frac {1}{\sqrt {.3^{2}+.1^{2}}}}{\begin{bmatrix}.3&.1\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}3&1\end{bmatrix}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {.25^{2}+.75^{2}}}}{\begin{bmatrix}-.25&.75\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1&3\end{bmatrix}},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <msup> <mn> .25 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> <mo> + </mo> <msup> <mn> .75 </mn> <mrow class="MJX-TeXAtom-ORD"> <mn> 2 </mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <mn> .25 </mn> </mtd> <mtd> <mn> .75 </mn> </mtd> </mtr> </mtable> <mo> ] </mo> </mrow> </mrow> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <mn> 10 </mn> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> [ </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo> − </mo> <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 \mathbf {e} _{2}={\frac {1}{\sqrt {.25^{2}+.75^{2}}}}{\begin{bmatrix}-.25&.75\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1&3\end{bmatrix}},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f686d86250121661936ae156d34d9980dbb84679" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:48.893ex; height:6.509ex;" alt="{\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {.25^{2}+.75^{2}}}}{\begin{bmatrix}-.25&.75\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1&3\end{bmatrix}},}"> </noscript><span class="lazy-image-placeholder" style="width: 48.893ex;height: 6.509ex;vertical-align: -3.171ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f686d86250121661936ae156d34d9980dbb84679" data-alt="{\displaystyle \mathbf {e} _{2}={\frac {1}{\sqrt {.25^{2}+.75^{2}}}}{\begin{bmatrix}-.25&.75\end{bmatrix}}={\frac {1}{\sqrt {10}}}{\begin{bmatrix}-1&3\end{bmatrix}},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  as in the example above. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(7)"> <h2 id="Determinant_formula">Determinant formula</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=8&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Determinant formula" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-7 collapsible-block" id="mf-section-7"> The result of the Gram–Schmidt process may be expressed in a non-recursive formula using <a href="https://en-m-wikipedia-org.translate.goog/wiki/Determinant?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Determinant">determinants</a>. <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {e} _{j}={\frac {1}{\sqrt {D_{j-1}D_{j}}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> e </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msqrt> <msub> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> <msub> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </msqrt> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <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> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </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> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo> ⋯ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <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 fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </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> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo> ⋯ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </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 fence="false" stretchy="false"> ⟩ </mo> </mtd> </mtr> <mtr> <mtd> <mo> ⋮ </mo> </mtd> <mtd> <mo> ⋮ </mo> </mtd> <mtd> <mo> ⋱ </mo> </mtd> <mtd> <mo> ⋮ </mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <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"> <mi> j </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </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"> <mi> j </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo> ⋯ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> </mtr> <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> j </mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {e} _{j}={\frac {1}{\sqrt {D_{j-1}D_{j}}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b11867d999f8c1cdd75d9cfea32f9792cd016684" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:57.198ex; height:18.509ex;" alt="{\displaystyle \mathbf {e} _{j}={\frac {1}{\sqrt {D_{j-1}D_{j}}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 57.198ex;height: 18.509ex;vertical-align: -8.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b11867d999f8c1cdd75d9cfea32f9792cd016684" data-alt="{\displaystyle \mathbf {e} _{j}={\frac {1}{\sqrt {D_{j-1}D_{j}}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{j}={\frac {1}{D_{j-1}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}}"><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"> <mi> j </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn> 1 </mn> <msub> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <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> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </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> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo> ⋯ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <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 fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </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> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo> ⋯ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </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 fence="false" stretchy="false"> ⟩ </mo> </mtd> </mtr> <mtr> <mtd> <mo> ⋮ </mo> </mtd> <mtd> <mo> ⋮ </mo> </mtd> <mtd> <mo> ⋱ </mo> </mtd> <mtd> <mo> ⋮ </mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <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"> <mi> j </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </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"> <mi> j </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo> ⋯ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> </mtr> <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> j </mi> </mrow> </msub> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{j}={\frac {1}{D_{j-1}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/613d2dfdeb2576c4edfb7ab6c91079c80272ef4a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:52.3ex; height:18.509ex;" alt="{\displaystyle \mathbf {u} _{j}={\frac {1}{D_{j-1}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}}"> </noscript><span class="lazy-image-placeholder" style="width: 52.3ex;height: 18.509ex;vertical-align: -8.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/613d2dfdeb2576c4edfb7ab6c91079c80272ef4a" data-alt="{\displaystyle \mathbf {u} _{j}={\frac {1}{D_{j-1}}}{\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j-1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j-1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j-1}\rangle \\\mathbf {v} _{1}&\mathbf {v} _{2}&\cdots &\mathbf {v} _{j}\end{vmatrix}}}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{0}=1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mn> 0 </mn> </mrow> </msub> <mo> = </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle D_{0}=1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fdc1d6c1abc3149c8b83f61412a3c2c88e46e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.239ex; height:2.509ex;" alt="{\displaystyle D_{0}=1}"> </noscript><span class="lazy-image-placeholder" style="width: 7.239ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b7fdc1d6c1abc3149c8b83f61412a3c2c88e46e5" data-alt="{\displaystyle D_{0}=1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  and, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j\geq 1}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> j </mi> <mo> ≥ </mo> <mn> 1 </mn> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle j\geq 1} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620328020a958825ea8d3b814be5e3ed924203e5" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:5.246ex; height:2.509ex;" alt="{\displaystyle j\geq 1}"> </noscript><span class="lazy-image-placeholder" style="width: 5.246ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/620328020a958825ea8d3b814be5e3ed924203e5" data-alt="{\displaystyle j\geq 1}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> , <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{j}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle D_{j}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41eafa45dbe60f5e03b0281cfef1c2eca6bdc4d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.834ex; height:2.843ex;" alt="{\displaystyle D_{j}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.834ex;height: 2.843ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41eafa45dbe60f5e03b0281cfef1c2eca6bdc4d2" data-alt="{\displaystyle D_{j}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram_determinant?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Gram determinant">Gram determinant</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D_{j}={\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j}\rangle \end{vmatrix}}.}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> D </mi> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo> | </mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <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> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </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> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo> ⋯ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <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 fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </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> 2 </mn> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo> ⋯ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </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 fence="false" stretchy="false"> ⟩ </mo> </mtd> </mtr> <mtr> <mtd> <mo> ⋮ </mo> </mtd> <mtd> <mo> ⋮ </mo> </mtd> <mtd> <mo> ⋱ </mo> </mtd> <mtd> <mo> ⋮ </mo> </mtd> </mtr> <mtr> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <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"> <mi> j </mi> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </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"> <mi> j </mi> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> <mtd> <mo> ⋯ </mo> </mtd> <mtd> <mo fence="false" stretchy="false"> ⟨ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo> , </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo fence="false" stretchy="false"> ⟩ </mo> </mtd> </mtr> </mtable> <mo> | </mo> </mrow> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle D_{j}={\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j}\rangle \end{vmatrix}}.} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6979160bc14eb9126464787d668887646fd034c7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:41.747ex; height:14.843ex;" alt="{\displaystyle D_{j}={\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j}\rangle \end{vmatrix}}.}"> </noscript><span class="lazy-image-placeholder" style="width: 41.747ex;height: 14.843ex;vertical-align: -6.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6979160bc14eb9126464787d668887646fd034c7" data-alt="{\displaystyle D_{j}={\begin{vmatrix}\langle \mathbf {v} _{1},\mathbf {v} _{1}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{1}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{1}\rangle \\\langle \mathbf {v} _{1},\mathbf {v} _{2}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{2}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{2}\rangle \\\vdots &\vdots &\ddots &\vdots \\\langle \mathbf {v} _{1},\mathbf {v} _{j}\rangle &\langle \mathbf {v} _{2},\mathbf {v} _{j}\rangle &\cdots &\langle \mathbf {v} _{j},\mathbf {v} _{j}\rangle \end{vmatrix}}.}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  Note that the expression for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{k}}"><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"> <mi> k </mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{k}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6db1dab97ca1af730a50c30421af8c3fd020c836" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.574ex; height:2.009ex;" alt="{\displaystyle \mathbf {u} _{k}}"> </noscript><span class="lazy-image-placeholder" style="width: 2.574ex;height: 2.009ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6db1dab97ca1af730a50c30421af8c3fd020c836" data-alt="{\displaystyle \mathbf {u} _{k}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is a "formal" determinant, i.e. the matrix contains both scalars and vectors; the meaning of this expression is defined to be the result of a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Laplace_expansion?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Laplace expansion">cofactor expansion</a> along the row of vectors. The determinant formula for the Gram-Schmidt is computationally (exponentially) slower than the recursive algorithms described above; it is mainly of theoretical interest. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(8)"> <h2 id="Expressed_using_geometric_algebra">Expressed using geometric algebra</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=9&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Expressed using geometric algebra" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-8 collapsible-block" id="mf-section-8"> Expressed using notation used in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Geometric_algebra?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Geometric algebra">geometric algebra</a>, the unnormalized results of the Gram–Schmidt process can be expressed as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\sum _{j=1}^{k-1}(\mathbf {v} _{k}\cdot \mathbf {u} _{j})\mathbf {u} _{j}^{-1}\ ,}"><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"> <mi> k </mi> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </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> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </munderover> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> ⋅ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> </msub> <mo stretchy="false"> ) </mo> <msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> j </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msubsup> <mtext>   </mtext> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\sum _{j=1}^{k-1}(\mathbf {v} _{k}\cdot \mathbf {u} _{j})\mathbf {u} _{j}^{-1}\ ,} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eac7d77d7e364f6a8a6571c19ef58a22e152bc2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:27.796ex; height:7.676ex;" alt="{\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\sum _{j=1}^{k-1}(\mathbf {v} _{k}\cdot \mathbf {u} _{j})\mathbf {u} _{j}^{-1}\ ,}"> </noscript><span class="lazy-image-placeholder" style="width: 27.796ex;height: 7.676ex;vertical-align: -3.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eac7d77d7e364f6a8a6571c19ef58a22e152bc2" data-alt="{\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}-\sum _{j=1}^{k-1}(\mathbf {v} _{k}\cdot \mathbf {u} _{j})\mathbf {u} _{j}^{-1}\ ,}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  which is equivalent to the expression using the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {proj} }"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> proj </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {proj} } </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b381dfff8bb6455d984951bfa7879450806423ff" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.078ex; height:2.509ex;" alt="{\displaystyle \operatorname {proj} }"> </noscript><span class="lazy-image-placeholder" style="width: 4.078ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b381dfff8bb6455d984951bfa7879450806423ff" data-alt="{\displaystyle \operatorname {proj} }" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  operator defined above. The results can equivalently be expressed as<a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-5">[4]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}\wedge \mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1}(\mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1})^{-1},}"><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"> <mi> k </mi> </mrow> </msub> <mo> = </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> </mrow> </msub> <mo> ∧ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> <mo> ∧ </mo> <mo> ⋅ </mo> <mo> ⋅ </mo> <mo> ⋅ </mo> <mo> ∧ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <mo stretchy="false"> ( </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi> k </mi> <mo> − </mo> <mn> 1 </mn> </mrow> </msub> <mo> ∧ </mo> <mo> ⋅ </mo> <mo> ⋅ </mo> <mo> ⋅ </mo> <mo> ∧ </mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn> 1 </mn> </mrow> </msub> <msup> <mo stretchy="false"> ) </mo> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}\wedge \mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1}(\mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1})^{-1},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0357885e1926f46e807c3f334bf253f31e7d25a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:45.951ex; height:3.176ex;" alt="{\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}\wedge \mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1}(\mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1})^{-1},}"> </noscript><span class="lazy-image-placeholder" style="width: 45.951ex;height: 3.176ex;vertical-align: -0.838ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f0357885e1926f46e807c3f334bf253f31e7d25a" data-alt="{\displaystyle \mathbf {u} _{k}=\mathbf {v} _{k}\wedge \mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1}(\mathbf {v} _{k-1}\wedge \cdot \cdot \cdot \wedge \mathbf {v} _{1})^{-1},}" data-class="mwe-math-fallback-image-display mw-invert skin-invert">  which is closely related to the expression using determinants above. </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(9)"> <h2 id="Alternatives">Alternatives</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=10&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Alternatives" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-9 collapsible-block" id="mf-section-9"> Other <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonalization?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthogonalization">orthogonalization</a> algorithms use <a href="https://en-m-wikipedia-org.translate.goog/wiki/Householder_transformation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Householder transformation">Householder transformations</a> or <a href="https://en-m-wikipedia-org.translate.goog/wiki/Givens_rotation?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Givens rotation">Givens rotations</a>. The algorithms using Householder transformations are more stable than the stabilized Gram–Schmidt process. On the other hand, the Gram–Schmidt process produces the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> j </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle j} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"> </noscript><span class="lazy-image-placeholder" style="width: 0.985ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" data-alt="{\displaystyle j}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> th orthogonalized vector after the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle j}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> j </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle j} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.027ex; width:0.985ex; height:2.509ex;" alt="{\displaystyle j}"> </noscript><span class="lazy-image-placeholder" style="width: 0.985ex;height: 2.509ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f461e54f5c093e92a55547b9764291390f0b5d0" data-alt="{\displaystyle j}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> th iteration, while orthogonalization using <a href="https://en-m-wikipedia-org.translate.goog/wiki/Householder_reflection?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Householder reflection">Householder reflections</a> produces all the vectors only at the end. This makes only the Gram–Schmidt process applicable for <a href="https://en-m-wikipedia-org.translate.goog/wiki/Iterative_method?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Iterative method">iterative methods</a> like the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Arnoldi_iteration?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Arnoldi iteration">Arnoldi iteration</a>. Yet another alternative is motivated by the use of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cholesky_decomposition?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cholesky decomposition">Cholesky decomposition</a> for <a href="https://en-m-wikipedia-org.translate.goog/wiki/Ordinary_least_squares?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Ordinary least squares">inverting the matrix of the normal equations in linear least squares</a>. Let <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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.787ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" data-alt="{\displaystyle V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  be a <a href="https://en-m-wikipedia-org.translate.goog/wiki/Full_rank?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Full rank">full column rank</a> matrix, whose columns need to be orthogonalized. The matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> ∗ </mo> </mrow> </msup> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V^{*}V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5cdcfc5927ea0468612f7735cdfce9e9ba05d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.758ex; height:2.343ex;" alt="{\displaystyle V^{*}V}"> </noscript><span class="lazy-image-placeholder" style="width: 4.758ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5cdcfc5927ea0468612f7735cdfce9e9ba05d7" data-alt="{\displaystyle V^{*}V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  is <a href="https://en-m-wikipedia-org.translate.goog/wiki/Hermitian_matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Hermitian matrix">Hermitian</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Positive_definite_matrix?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Positive definite matrix">positive definite</a>, so it can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}V=LL^{*},}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> ∗ </mo> </mrow> </msup> <mi> V </mi> <mo> = </mo> <mi> L </mi> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> ∗ </mo> </mrow> </msup> <mo> , </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V^{*}V=LL^{*},} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7497cc68a2a14ba2320b94c6c2c26441c114a0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.724ex; height:2.676ex;" alt="{\displaystyle V^{*}V=LL^{*},}"> </noscript><span class="lazy-image-placeholder" style="width: 12.724ex;height: 2.676ex;vertical-align: -0.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8d7497cc68a2a14ba2320b94c6c2c26441c114a0" data-alt="{\displaystyle V^{*}V=LL^{*},}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  using the <a href="https://en-m-wikipedia-org.translate.goog/wiki/Cholesky_decomposition?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Cholesky decomposition">Cholesky decomposition</a>. The lower triangular matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> L </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle L} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}"> </noscript><span class="lazy-image-placeholder" style="width: 1.583ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" data-alt="{\displaystyle L}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  with strictly positive diagonal entries is <a href="https://en-m-wikipedia-org.translate.goog/wiki/Invertible?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Invertible">invertible</a>. Then columns of the matrix <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U=V\left(L^{-1}\right)^{*}}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi> U </mi> <mo> = </mo> <mi> V </mi> <msup> <mrow> <mo> ( </mo> <msup> <mi> L </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> − </mo> <mn> 1 </mn> </mrow> </msup> <mo> ) </mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo> ∗ </mo> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle U=V\left(L^{-1}\right)^{*}} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e93e425160d3dd418befc5c2c6772d5bd0e81c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:13.768ex; height:3.509ex;" alt="{\displaystyle U=V\left(L^{-1}\right)^{*}}"> </noscript><span class="lazy-image-placeholder" style="width: 13.768ex;height: 3.509ex;vertical-align: -1.005ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38e93e425160d3dd418befc5c2c6772d5bd0e81c" data-alt="{\displaystyle U=V\left(L^{-1}\right)^{*}}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  are <a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthonormal?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Orthonormal">orthonormal</a> and <a href="https://en-m-wikipedia-org.translate.goog/wiki/Linear_span?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Linear span">span</a> the same subspace as the columns of the original 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> <noscript> <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}"> </noscript><span class="lazy-image-placeholder" style="width: 1.787ex;height: 2.176ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/af0f6064540e84211d0ffe4dac72098adfa52845" data-alt="{\displaystyle V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert"> . The explicit use of the product <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V^{*}V}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi> V </mi> <mrow class="MJX-TeXAtom-ORD"> <mo> ∗ </mo> </mrow> </msup> <mi> V </mi> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle V^{*}V} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5cdcfc5927ea0468612f7735cdfce9e9ba05d7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.758ex; height:2.343ex;" alt="{\displaystyle V^{*}V}"> </noscript><span class="lazy-image-placeholder" style="width: 4.758ex;height: 2.343ex;vertical-align: -0.338ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7c5cdcfc5927ea0468612f7735cdfce9e9ba05d7" data-alt="{\displaystyle V^{*}V}" data-class="mwe-math-fallback-image-inline mw-invert skin-invert">  makes the algorithm unstable, especially if the product's <a href="https://en-m-wikipedia-org.translate.goog/wiki/Condition_number?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Condition number">condition number</a> is large. Nevertheless, this algorithm is used in practice and implemented in some software packages because of its high efficiency and simplicity. In <a href="https://en-m-wikipedia-org.translate.goog/wiki/Quantum_mechanics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Quantum mechanics">quantum mechanics</a> there are several orthogonalization schemes with characteristics better suited for certain applications than original Gram–Schmidt. Nevertheless, it remains a popular and effective algorithm for even the largest electronic structure calculations.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-6">[5]</a> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(10)"> <h2 id="Run-time_complexity">Run-time complexity</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=11&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Run-time complexity" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-10 collapsible-block" id="mf-section-10"> Gram-Schmidt orthogonalization can be done in <a href="https://en-m-wikipedia-org.translate.goog/wiki/Strongly-polynomial_time?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Strongly-polynomial time">strongly-polynomial time</a>. The run-time analysis is similar to that of <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gaussian_elimination?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gaussian elimination">Gaussian elimination</a>.<a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_note-:0-7">[6]</a>: 40 </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(11)"> <h2 id="See_also">See also</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=12&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: See also" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-11 collapsible-block" id="mf-section-11"> <ul> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Linear_algebra?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Linear algebra">Linear algebra</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Recursion?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Recursion">Recursion</a></li> <li><a href="https://en-m-wikipedia-org.translate.goog/wiki/Orthogonality_(mathematics)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Orthogonality (mathematics)">Orthogonality (mathematics)</a></li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(12)"> <h2 id="References">References</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=13&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: References" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-12 collapsible-block" id="mf-section-12"> <div class="mw-references-wrap"> <ol class="references"> <li id="cite_note-1"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-1">^</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="CITEREFCheneyKincaid2009" class="citation book cs1">Cheney, Ward; Kincaid, David (2009). <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DGg3Uj1GkHK8C%26pg%3DPA544">Linear Algebra: Theory and Applications</a>. Sudbury, Ma: Jones and Bartlett. pp. 544, 558. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-7637-5020-6?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-7637-5020-6"><bdi>978-0-7637-5020-6</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%3A+Theory+and+Applications&rft.place=Sudbury%2C+Ma&rft.pages=544%2C+558&rft.pub=Jones+and+Bartlett&rft.date=2009&rft.isbn=978-0-7637-5020-6&rft.aulast=Cheney&rft.aufirst=Ward&rft.au=Kincaid%2C+David&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DGg3Uj1GkHK8C%26pg%3DPA544&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGram%E2%80%93Schmidt+process" class="Z3988"></li> <li id="cite_note-FOOTNOTEGolubVan_Loan1996§5.2.8-3"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-FOOTNOTEGolubVan_Loan1996%C2%A75.2.8_3-0">^</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#CITEREFGolubVan_Loan1996">Golub & Van Loan 1996</a>, §5.2.8.</li> <li id="cite_note-4"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPursellTrimble1991" class="citation journal cs1">Pursell, Lyle; Trimble, S. 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(2011). "First-principles calculations of electron states of a silicon nanowire with 100,000 atoms on the K computer". Proceedings of 2011 International Conference for High Performance Computing, Networking, Storage and Analysis. pp. 1:1–1:11. <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1145%252F2063384.2063386">10.1145/2063384.2063386</a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/9781450307710?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/9781450307710"><bdi>9781450307710</bdi></a>. <a href="https://en-m-wikipedia-org.translate.goog/wiki/S2CID_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://api.semanticscholar.org/CorpusID:14316074">14316074</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=First-principles+calculations+of+electron+states+of+a+silicon+nanowire+with+100%2C000+atoms+on+the+K+computer&rft.btitle=Proceedings+of+2011+International+Conference+for+High+Performance+Computing%2C+Networking%2C+Storage+and+Analysis&rft.pages=1%3A1-1%3A11&rft.date=2011&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A14316074%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1145%2F2063384.2063386&rft.isbn=9781450307710&rft.aulast=Pursell&rft.aufirst=Yukihiro&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGram%E2%80%93Schmidt+process" class="Z3988"></li> <li id="cite_note-:0-7"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-:0_7-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrötschelLovászSchrijver1993" class="citation cs2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Martin_Gr%C3%B6tschel?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Martin Grötschel">Grötschel, Martin</a>; <a href="https://en-m-wikipedia-org.translate.goog/wiki/L%C3%A1szl%C3%B3_Lov%C3%A1sz?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="László Lovász">Lovász, László</a>; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Alexander_Schrijver?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Alexander Schrijver">Schrijver, Alexander</a> (1993), <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://books.google.com/books?id%3DhWvmCAAAQBAJ%26pg%3DPA281">Geometric algorithms and combinatorial optimization</a>, Algorithms and Combinatorics, vol. 2 (2nd ed.), Springer-Verlag, Berlin, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Doi_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://doi.org/10.1007%252F978-3-642-78240-4">10.1007/978-3-642-78240-4</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-3-642-78242-8?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-3-642-78242-8"><bdi>978-3-642-78242-8</bdi></a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/MR_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://mathscinet.ams.org/mathscinet-getitem?mr%3D1261419">1261419</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Geometric+algorithms+and+combinatorial+optimization&rft.series=Algorithms+and+Combinatorics&rft.edition=2nd&rft.pub=Springer-Verlag%2C+Berlin&rft.date=1993&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D1261419%23id-name%3DMR&rft_id=info%3Adoi%2F10.1007%2F978-3-642-78240-4&rft.isbn=978-3-642-78242-8&rft.aulast=Gr%C3%B6tschel&rft.aufirst=Martin&rft.au=Lov%C3%A1sz%2C+L%C3%A1szl%C3%B3&rft.au=Schrijver%2C+Alexander&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DhWvmCAAAQBAJ%26pg%3DPA281&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGram%E2%80%93Schmidt+process" class="Z3988"></li> </ol> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(13)"> <h2 id="Notes">Notes</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=14&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Notes" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-13 collapsible-block" id="mf-section-13"> <div class="mw-references-wrap"> <ol class="references"> <li id="cite_note-2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gram%E2%80%93Schmidt_process?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB#cite_ref-2">^</a> In the complex case, this assumes that the inner product is linear in the first argument and conjugate-linear in the second. In physics a more common convention is linearity in the second argument, in which case we define <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} .}"><semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi> proj </mi> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> </mrow> </msub> <mo> ⁡ </mo> <mo stretchy="false"> ( </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> v </mi> </mrow> <mo stretchy="false"> ) </mo> <mo> = </mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mo fence="false" 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 fence="false" stretchy="false"> ⟩ </mo> </mrow> <mrow> <mo fence="false" stretchy="false"> ⟨ </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo> , </mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo fence="false" stretchy="false"> ⟩ </mo> </mrow> </mfrac> </mrow> <mspace width="thinmathspace"></mspace> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold"> u </mi> </mrow> <mo> . </mo> </mstyle> </mrow> <annotation encoding="application/x-tex"> {\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} .} </annotation> </semantics> </math> <noscript> <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8caeff704bea6c5baa95acc31508776cd9543e94" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:20.849ex; height:6.509ex;" alt="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} .}"> </noscript><span class="lazy-image-placeholder" style="width: 20.849ex;height: 6.509ex;vertical-align: -2.671ex;" data-src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8caeff704bea6c5baa95acc31508776cd9543e94" data-alt="{\displaystyle \operatorname {proj} _{\mathbf {u} }(\mathbf {v} )={\frac {\langle \mathbf {u} ,\mathbf {v} \rangle }{\langle \mathbf {u} ,\mathbf {u} \rangle }}\,\mathbf {u} .}" data-class="mwe-math-fallback-image-display mw-invert skin-invert"> </li> </ol> </div> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(14)"> <h2 id="Sources">Sources</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=15&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: Sources" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-14 collapsible-block" id="mf-section-14"> <ul> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBau_IIITrefethen1997" class="citation cs2">Bau III, David; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Lloyd_N._Trefethen?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="Lloyd N. Trefethen">Trefethen, Lloyd N.</a> (1997), Numerical linear algebra, Philadelphia: Society for Industrial and Applied Mathematics, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-89871-361-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-89871-361-9"><bdi>978-0-89871-361-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Numerical+linear+algebra&rft.place=Philadelphia&rft.pub=Society+for+Industrial+and+Applied+Mathematics&rft.date=1997&rft.isbn=978-0-89871-361-9&rft.aulast=Bau+III&rft.aufirst=David&rft.au=Trefethen%2C+Lloyd+N.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGram%E2%80%93Schmidt+process" class="Z3988">.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGolubVan_Loan1996" class="citation cs2"><a href="https://en-m-wikipedia-org.translate.goog/wiki/Gene_H._Golub?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Gene H. Golub">Golub, Gene H.</a>; <a href="https://en-m-wikipedia-org.translate.goog/wiki/Charles_F._Van_Loan?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Charles F. Van Loan">Van Loan, Charles F.</a> (1996), Matrix Computations (3rd ed.), Johns Hopkins, <a href="https://en-m-wikipedia-org.translate.goog/wiki/ISBN_(identifier)?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="https://en-m-wikipedia-org.translate.goog/wiki/Special:BookSources/978-0-8018-5414-9?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Special:BookSources/978-0-8018-5414-9"><bdi>978-0-8018-5414-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Matrix+Computations&rft.edition=3rd&rft.pub=Johns+Hopkins&rft.date=1996&rft.isbn=978-0-8018-5414-9&rft.aulast=Golub&rft.aufirst=Gene+H.&rft.au=Van+Loan%2C+Charles+F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGram%E2%80%93Schmidt+process" class="Z3988">.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreub1975" class="citation cs2">Greub, Werner (1975), Linear Algebra (4th ed.), Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+Algebra&rft.edition=4th&rft.pub=Springer&rft.date=1975&rft.aulast=Greub&rft.aufirst=Werner&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGram%E2%80%93Schmidt+process" class="Z3988">.</li> <li> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSoliverezGagliano1985" class="citation cs2">Soliverez, C. E.; Gagliano, E. (1985), <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20140307095009/http://rmf.smf.mx/pdf/rmf/31/4/31_4_743.pdf">"Orthonormalization on the plane: a geometric approach"</a> (PDF), Mex. J. Phys., 31 (4): 743–758, archived from <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://rmf.smf.mx/pdf/rmf/31/4/31_4_743.pdf">the original</a> (PDF) on 2014-03-07, retrieved 2013-06-22</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mex.+J.+Phys.&rft.atitle=Orthonormalization+on+the+plane%3A+a+geometric+approach&rft.volume=31&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E743-%3C%2Fspan%3E758&rft.date=1985&rft.aulast=Soliverez&rft.aufirst=C.+E.&rft.au=Gagliano%2C+E.&rft_id=http%3A%2F%2Frmf.smf.mx%2Fpdf%2Frmf%2F31%2F4%2F31_4_743.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGram%E2%80%93Schmidt+process" class="Z3988">.</li> </ul> </section> <div class="mw-heading mw-heading2 section-heading" onclick="mfTempOpenSection(15)"> <h2 id="External_links">External links</h2> <a role="button" href="https://en-m-wikipedia-org.translate.goog/w/index.php?title=Gram%E2%80%93Schmidt_process&action=edit&section=16&_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Edit section: External links" class="cdx-button cdx-button--size-large cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--icon-only cdx-button--weight-quiet "> edit </a> </div> <section class="mf-section-15 collapsible-block" id="mf-section-15"> <style data-mw-deduplicate="TemplateStyles:r1266661725">.mw-parser-output .portalbox{padding:0;margin:0.5em 0;display:table;box-sizing:border-box;max-width:175px;list-style:none}.mw-parser-output .portalborder{border:1px solid var(--border-color-base,#a2a9b1);padding:0.1em;background:var(--background-color-neutral-subtle,#f8f9fa)}.mw-parser-output .portalbox-entry{display:table-row;font-size:85%;line-height:110%;height:1.9em;font-style:italic;font-weight:bold}.mw-parser-output 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href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.encyclopediaofmath.org/index.php?title%3DOrthogonalization">"Orthogonalization"</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/Encyclopedia_of_Mathematics?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="Encyclopedia of Mathematics">Encyclopedia of Mathematics</a>, <a href="https://en-m-wikipedia-org.translate.goog/wiki/European_Mathematical_Society?_x_tr_sl=auto&_x_tr_tl=en&_x_tr_hl=en-GB" title="European Mathematical Society">EMS Press</a>, 2001 [1994]</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Orthogonalization&rft.btitle=Encyclopedia+of+Mathematics&rft.pub=EMS+Press&rft.date=2001&rft_id=https%3A%2F%2Fwww.encyclopediaofmath.org%2Findex.php%3Ftitle%3DOrthogonalization&rfr_id=info%3Asid%2Fen.wikipedia.org%3AGram%E2%80%93Schmidt+process" class="Z3988"></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://web.archive.org/web/20160402140129/https://www.math.hmc.edu/calculus/tutorials/gramschmidt/gramschmidt.pdf">Harvey Mudd College Math Tutorial on the Gram-Schmidt algorithm</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://jeff560.tripod.com/g.html">Earliest known uses of some of the words of mathematics: G</a> The entry "Gram-Schmidt orthogonalization" has some information and references on the origins of the method.</li> <li>Demos: <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bigsigma.com/linear-algebra/gram-schmidt-process/%23plain">Gram Schmidt process in plane</a> and <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://bigsigma.com/linear-algebra/gram-schmidt-process/%23space">Gram Schmidt process in space</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://www.math.ucla.edu/~tao/resource/general/115a.3.02f/GramSchmidt.html">Gram-Schmidt orthogonalization applet</a></li> <li><a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=http://www.nag.co.uk/numeric/fl/nagdoc_fl24/html/F05/f05conts.html">NAG Gram–Schmidt orthogonalization of n vectors of order m routine</a></li> <li>Proof: <a rel="nofollow" class="external text" href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://planetmath.org/ProofOfGramSchmidtOrthogonalizationProcedure">Raymond Puzio, Keenan Kidwell. "proof of Gram-Schmidt orthogonalization algorithm" (version 8). 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href="https://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sr.wikipedia.org/wiki/%25D0%2593%25D1%2580%25D0%25B0%25D0%25BC%25E2%2580%2594%25D0%25A8%25D0%25BC%25D0%25B8%25D1%2582%25D0%25BE%25D0%25B2_%25D0%25BF%25D0%25BE%25D1%2581%25D1%2582%25D1%2583%25D0%25BF%25D0%25B0%25D0%25BA_%25D0%25BE%25D1%2580%25D1%2582%25D0%25BE%25D0%25BD%25D0%25BE%25D1%2580%25D0%25BC%25D0%25B0%25D0%25BB%25D0%25B8%25D0%25B7%25D0%25B0%25D1%2586%25D0%25B8%25D1%2598%25D0%25B5" title="Грам—Шмитов поступак ортонормализације – Serbian" lang="sr" hreflang="sr" data-title="Грам—Шмитов поступак ортонормализације" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sh.wikipedia.org/wiki/Gram%25E2%2580%2593Schmidtov_postupak" title="Gram–Schmidtov postupak – Serbo-Croatian" lang="sh" hreflang="sh" data-title="Gram–Schmidtov postupak" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://fi.wikipedia.org/wiki/Gramin%25E2%2580%2593Schmidtin_ortogonalisoimismenetelm%25C3%25A4" title="Gramin–Schmidtin ortogonalisoimismenetelmä – Finnish" lang="fi" hreflang="fi" data-title="Gramin–Schmidtin ortogonalisoimismenetelmä" 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://translate.google.com/website?sl=auto&tl=en&hl=en-GB&u=https://sv.wikipedia.org/wiki/Gram%25E2%2580%2593Schmidts_ortogonaliseringsprocess" title="Gram–Schmidts ortogonaliseringsprocess – Swedish" lang="sv" hreflang="sv" 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Gram–Schmidt process - Wikipedia