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href="https://de.wikipedia.org/wiki/Ganzzahlige_unimodulare_Matrix" title="Ganzzahlige unimodulare Matrix – German" lang="de" hreflang="de" data-title="Ganzzahlige unimodulare Matrix" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%A0%CE%B9%CE%BD%CE%AC%CE%BA%CE%B1%CF%82_%CE%BC%CE%B5_%CE%BF%CF%81%CE%AF%CE%B6%CE%BF%CF%85%CF%83%CE%B1_%CE%BC%CE%BF%CE%BD%CE%AC%CE%B4%CE%B1" title="Πινάκας με ορίζουσα μονάδα – Greek" lang="el" hreflang="el" data-title="Πινάκας με ορίζουσα μονάδα" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Matriz_unimodular" title="Matriz unimodular – Spanish" lang="es" hreflang="es" data-title="Matriz unimodular" 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data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">This article is about matrices whose entries are <a href="/wiki/Integer_number" class="mw-redirect" title="Integer number">integer numbers</a>. For use of term <b>unimodular</b> in connection with <a href="/wiki/Polynomial_matrix" title="Polynomial matrix">polynomial matrices</a>, see <a href="/wiki/Unimodular_polynomial_matrix" title="Unimodular polynomial matrix">Unimodular polynomial matrix</a>.</div> <p class="mw-empty-elt"> </p> <div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Integer matrices with +1 or -1 determinant; invertible over the integers. GL_n(Z)</div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>unimodular matrix</b> <i>M</i> is a <a href="/wiki/Square_matrix" title="Square matrix">square</a> <a href="/wiki/Integer_matrix" title="Integer matrix">integer matrix</a> having <a href="/wiki/Determinant" title="Determinant">determinant</a> +1 or −1. Equivalently, it is an integer matrix that is <a href="/wiki/Invertible_matrix" title="Invertible matrix">invertible</a> over the <a href="/wiki/Integer" title="Integer">integers</a>: there is an integer matrix <i>N</i> that is its inverse (these are equivalent under <a href="/wiki/Cramer%27s_rule" title="Cramer's rule">Cramer's rule</a>). Thus every equation <span class="nowrap"><i>Mx</i> = <i>b</i></span>, where <i>M</i> and <i>b</i> both have integer components and <i>M</i> is unimodular, has an integer solution. The <i>n</i> × <i>n</i> unimodular matrices form a <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> called the <i>n</i> × <i>n</i> <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> over <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbb {Z} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {Z} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/449494a083e0a1fda2b61c62b2f09b6bee4633dc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.55ex; height:2.176ex;" alt="{\displaystyle \mathbb {Z} }"></span>, which is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(\mathbb {Z} )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(\mathbb {Z} )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1318c85c8812d864d6226f1000ca8881a6b864bf" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.855ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(\mathbb {Z} )}"></span>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples_of_unimodular_matrices">Examples of unimodular matrices</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unimodular_matrix&action=edit&section=1" title="Edit section: Examples of unimodular matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Unimodular matrices form a <a href="/wiki/Subgroup" title="Subgroup">subgroup</a> of the <a href="/wiki/General_linear_group" title="General linear group">general linear group</a> under <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix multiplication</a>, i.e. the following matrices are unimodular: </p> <ul><li><a href="/wiki/Identity_matrix" title="Identity matrix">Identity matrix</a></li> <li>The <a href="/wiki/Matrix_inverse" class="mw-redirect" title="Matrix inverse">inverse</a> of a unimodular matrix</li> <li>The <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">product</a> of two unimodular matrices</li></ul> <p>Other examples include: </p> <ul><li><a href="/wiki/Pascal_matrix" title="Pascal matrix">Pascal matrices</a></li> <li><a href="/wiki/Permutation_matrix" title="Permutation matrix">Permutation matrices</a></li> <li>the three transformation matrices in the ternary <a href="/wiki/Tree_of_primitive_Pythagorean_triples" title="Tree of primitive Pythagorean triples">tree of primitive Pythagorean triples</a></li> <li>Certain transformation matrices for <a href="/wiki/Rotation_matrix" title="Rotation matrix">rotation</a>, <a href="/wiki/Shear_mapping" title="Shear mapping">shearing</a> (both with determinant 1) and <a href="/wiki/Reflection_matrix" class="mw-redirect" title="Reflection matrix">reflection</a> (determinant −1).</li> <li>The unimodular matrix used (possibly implicitly) in <a href="/wiki/Lattice_reduction" title="Lattice reduction">lattice reduction</a> and in the <a href="/wiki/Hermite_normal_form" title="Hermite normal form">Hermite normal form</a> of matrices.</li> <li>The <a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a> of two unimodular matrices is also unimodular. This follows since <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \det(A\otimes B)=(\det A)^{q}(\det B)^{p},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo movablelimits="true" form="prefix">det</mo> <mo stretchy="false">(</mo> <mi>A</mi> <mo>⊗</mo> <mi>B</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">det</mo> <mi>A</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>q</mi> </mrow> </msup> <mo stretchy="false">(</mo> <mo movablelimits="true" form="prefix">det</mo> <mi>B</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>p</mi> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \det(A\otimes B)=(\det A)^{q}(\det B)^{p},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ea2c398cb9c85b4f47a845fe34f5ee5e57b40e6d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:31.538ex; height:2.843ex;" alt="{\displaystyle \det(A\otimes B)=(\det A)^{q}(\det B)^{p},}"></span> where <i>p</i> and <i>q</i> are the dimensions of <i>A</i> and <i>B</i>, respectively.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Total_unimodularity">Total unimodularity</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unimodular_matrix&action=edit&section=2" title="Edit section: Total unimodularity"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A <b>totally unimodular matrix</b><sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> (TU matrix) is a matrix for which every square <a href="/wiki/Invertible_matrix" title="Invertible matrix">non-singular</a> <a href="/wiki/Submatrix" class="mw-redirect" title="Submatrix">submatrix</a> is unimodular. Equivalently, every square submatrix has determinant 0, +1 or −1. A totally unimodular matrix need not be square itself. From the definition it follows that any submatrix of a totally unimodular matrix is itself totally unimodular (TU). Furthermore it follows that any TU matrix has only 0, +1 or −1 entries. The <a href="/wiki/Converse_(logic)" title="Converse (logic)">converse</a> is not true, i.e., a matrix with only 0, +1 or −1 entries is not necessarily unimodular. A matrix is TU if and only if its <a href="/wiki/Transpose" title="Transpose">transpose</a> is TU. </p><p>Totally unimodular matrices are extremely important in <a href="/wiki/Polyhedral_combinatorics" title="Polyhedral combinatorics">polyhedral combinatorics</a> and <a href="/wiki/Combinatorial_optimization" title="Combinatorial optimization">combinatorial optimization</a> since they give a quick way to verify that a <a href="/wiki/Linear_program" class="mw-redirect" title="Linear program">linear program</a> is <a href="/wiki/Linear_programming#Integral_linear_programs" title="Linear programming">integral</a> (has an integral optimum, when any optimum exists). Specifically, if <i>A</i> is TU and <i>b</i> is integral, then linear programs of forms like <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\min c^{\top }x\mid Ax\geq b,x\geq 0\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo movablelimits="true" form="prefix">min</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi>x</mi> <mo>∣</mo> <mi>A</mi> <mi>x</mi> <mo>≥</mo> <mi>b</mi> <mo>,</mo> <mi>x</mi> <mo>≥</mo> <mn>0</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\min c^{\top }x\mid Ax\geq b,x\geq 0\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f441d7249eabe9736c521b5db44f99149efdf96d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.165ex; height:3.176ex;" alt="{\displaystyle \{\min c^{\top }x\mid Ax\geq b,x\geq 0\}}"></span> or <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\max c^{\top }x\mid Ax\leq b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mo movablelimits="true" form="prefix">max</mo> <msup> <mi>c</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">⊤</mi> </mrow> </msup> <mi>x</mi> <mo>∣</mo> <mi>A</mi> <mi>x</mi> <mo>≤</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{\max c^{\top }x\mid Ax\leq b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3196deb25aa0a2d7a175f67b56062b60472ff635" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.991ex; height:3.176ex;" alt="{\displaystyle \{\max c^{\top }x\mid Ax\leq b\}}"></span> have integral optima, for any <i>c</i>. Hence if <i>A</i> is totally unimodular and <i>b</i> is integral, every extreme point of the feasible region (e.g. <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{x\mid Ax\geq b\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mi>x</mi> <mo>∣</mo> <mi>A</mi> <mi>x</mi> <mo>≥</mo> <mi>b</mi> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{x\mid Ax\geq b\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01b1cfe1716946984ebc915a8c4b1468fa1649a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.761ex; height:2.843ex;" alt="{\displaystyle \{x\mid Ax\geq b\}}"></span>) is integral and thus the feasible region is an <a href="/wiki/Linear_programming#Integral_linear_programs" title="Linear programming">integral</a> polyhedron. </p> <div class="mw-heading mw-heading3"><h3 id="Common_totally_unimodular_matrices">Common totally unimodular matrices</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unimodular_matrix&action=edit&section=3" title="Edit section: Common totally unimodular matrices"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>1. The unoriented incidence matrix of a <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite graph</a>, which is the coefficient matrix for bipartite <a href="/wiki/Matching_(graph_theory)" title="Matching (graph theory)">matching</a>, is totally unimodular (TU). (The unoriented incidence matrix of a non-bipartite graph is not TU.) More generally, in the appendix to a paper by Heller and Tompkins,<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> A.J. Hoffman and D. Gale prove the following. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> be an <i>m</i> by <i>n</i> matrix whose rows can be partitioned into two <a href="/wiki/Disjoint_sets" title="Disjoint sets">disjoint sets</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>. Then the following four conditions together are <a href="/wiki/Necessary_and_sufficient_conditions" class="mw-redirect" title="Necessary and sufficient conditions">sufficient</a> for <i>A</i> to be totally unimodular: </p> <ul><li>Every entry in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is 0, +1, or −1;</li> <li>Every column of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> contains at most two non-zero (i.e., +1 or −1) entries;</li> <li>If two non-zero entries in a column of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> have the same sign, then the row of one is in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, and the other in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>;</li> <li>If two non-zero entries in a column of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> have opposite signs, then the rows of both are in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle B}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>B</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle B}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47136aad860d145f75f3eed3022df827cee94d7a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle B}"></span>, or both in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}"></span>.</li></ul> <p>It was realized later that these conditions define an incidence matrix of a balanced <a href="/wiki/Signed_graph#Incidence_matrix" title="Signed graph">signed graph</a>; thus, this example says that the incidence matrix of a signed graph is totally unimodular if the signed graph is balanced. The converse is valid for signed graphs without half edges (this generalizes the property of the unoriented incidence matrix of a graph).<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> </p><p>2. The <a href="/wiki/Constraint_(mathematics)" title="Constraint (mathematics)">constraints</a> of <a href="/wiki/Maximum_flow" class="mw-redirect" title="Maximum flow">maximum flow</a> and <a href="/wiki/Minimum_cost_flow" class="mw-redirect" title="Minimum cost flow">minimum cost flow</a> problems yield a coefficient matrix with these properties (and with empty <i>C</i>). Thus, such network flow problems with bounded integer capacities have an integral optimal value. Note that this does not apply to <a href="/wiki/Multi-commodity_flow_problem" title="Multi-commodity flow problem">multi-commodity flow problems</a>, in which it is possible to have fractional optimal value even with bounded integer capacities. </p><p>3. The consecutive-ones property: if <i>A</i> is (or can be permuted into) a 0-1 matrix in which for every row, the 1s appear consecutively, then <i>A</i> is TU. (The same holds for columns since the transpose of a TU matrix is also TU.) <sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>4. Every <b>network matrix</b> is TU. The rows of a network matrix correspond to a tree <span class="nowrap"><i>T</i> = (<i>V</i>, <i>R</i>)</span>, each of whose arcs has an arbitrary orientation (it is not necessary that there exist a root vertex <i>r</i> such that the tree is "rooted into <i>r</i>" or "out of <i>r</i>").The columns correspond to another set <i>C</i> of arcs on the same vertex set <i>V</i>. To compute the entry at row <i>R</i> and column <span class="nowrap"><i>C</i> = <i>st</i></span>, look at the <i>s</i>-to-<i>t</i> path <i>P</i> in <i>T</i>; then the entry is: </p> <ul><li>+1 if arc <i>R</i> appears forward in <i>P</i>,</li> <li>−1 if arc <i>R</i> appears backwards in <i>P</i>,</li> <li>0 if arc <i>R</i> does not appear in <i>P</i>.</li></ul> <p>See more in Schrijver (2003). </p><p>5. Ghouila-Houri showed that a matrix is TU iff for every subset <i>R</i> of rows, there is an assignment <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s:R\to \pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>s</mi> <mo>:</mo> <mi>R</mi> <mo stretchy="false">→</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle s:R\to \pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb379ec96165f22d73d003b93754ae30109ed137" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.376ex; height:2.176ex;" alt="{\displaystyle s:R\to \pm 1}"></span> of signs to rows so that the signed sum <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sum _{r\in R}s(r)r}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>r</mi> <mo>∈</mo> <mi>R</mi> </mrow> </munder> <mi>s</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi>r</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \sum _{r\in R}s(r)r}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4e4206068c98316855d8a88c92437435cb2a9457" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:8.739ex; height:5.676ex;" alt="{\displaystyle \sum _{r\in R}s(r)r}"></span> (which is a row vector of the same width as the matrix) has all its entries in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0,\pm 1\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo fence="false" stretchy="false">{</mo> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \{0,\pm 1\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/54d07858c9d2d2622a67ea6114fd7ca84c0f1e43" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.492ex; height:2.843ex;" alt="{\displaystyle \{0,\pm 1\}}"></span> (i.e. the row-submatrix has <a href="/wiki/Discrepancy_of_hypergraphs" title="Discrepancy of hypergraphs">discrepancy</a> at most one). This and several other if-and-only-if characterizations are proven in Schrijver (1998). </p><p>6. Hoffman and <a href="/wiki/Joseph_Kruskal" title="Joseph Kruskal">Kruskal</a><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> proved the following theorem. Suppose <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> is a <a href="/wiki/Directed_graph" title="Directed graph">directed graph</a> without 2-dicycles, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span> is the set of all <a href="/wiki/Dipath" class="mw-redirect" title="Dipath">dipaths</a> in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span>, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is the 0-1 incidence matrix of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V(G)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>V</mi> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V(G)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8ba39dee5fd7f4467e387af4026315fb1fb21628" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.423ex; height:2.843ex;" alt="{\displaystyle V(G)}"></span> versus <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>P</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle P}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b4dc73bf40314945ff376bd363916a738548d40a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}"></span>. Then <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span> is totally unimodular if and only if every simple arbitrarily-oriented cycle in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>G</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle G}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5f3c8921a3b352de45446a6789b104458c9f90b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}"></span> consists of alternating forwards and backwards arcs. </p><p>7. Suppose a matrix has 0-(<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pm }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>±</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \pm }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/869e366caf596564de4de06cb0ba124056d4064b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \pm }"></span>1) entries and in each column, the entries are non-decreasing from top to bottom (so all −1s are on top, then 0s, then 1s are on the bottom). Fujishige showed<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> that the matrix is TU iff every 2-by-2 submatrix has determinant in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0,\pm 1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mn>0</mn> <mo>,</mo> <mo>±</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 0,\pm 1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/392b6258d775add01c7092738b05926509fa7455" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.167ex; height:2.509ex;" alt="{\displaystyle 0,\pm 1}"></span>. </p><p>8. <a href="/wiki/Paul_Seymour_(mathematician)" title="Paul Seymour (mathematician)">Seymour</a> (1980)<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> proved a full characterization of all TU matrices, which we describe here only informally. Seymour's theorem is that a matrix is TU if and only if it is a certain natural combination of some <b>network matrices</b> and some copies of a particular 5-by-5 TU matrix. </p> <div class="mw-heading mw-heading3"><h3 id="Concrete_examples">Concrete examples</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unimodular_matrix&action=edit&section=4" title="Edit section: Concrete examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>1. The following matrix is totally unimodular: </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left[{\begin{array}{rrrrrr}-1&-1&0&0&0&+1\\+1&0&-1&-1&0&0\\0&+1&+1&0&-1&0\\0&0&0&+1&+1&-1\end{array}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right right right right right right" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>+</mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left[{\begin{array}{rrrrrr}-1&-1&0&0&0&+1\\+1&0&-1&-1&0&0\\0&+1&+1&0&-1&0\\0&0&0&+1&+1&-1\end{array}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60b3e8b8bd43e72651d7c5b4475c61e3ed049b4f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:39.164ex; height:12.509ex;" alt="{\displaystyle A=\left[{\begin{array}{rrrrrr}-1&-1&0&0&0&+1\\+1&0&-1&-1&0&0\\0&+1&+1&0&-1&0\\0&0&0&+1&+1&-1\end{array}}\right].}"></span></dd></dl> <p>This matrix arises as the coefficient matrix of the constraints in the linear programming formulation of the <a href="/wiki/Max-flow_min-cut_theorem" title="Max-flow min-cut theorem">maximum flow</a> problem on the following network: </p><p><span class="mw-default-size" typeof="mw:File"><a href="/wiki/File:Graph_for_example_adjacency_matrix.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Graph_for_example_adjacency_matrix.svg/701px-Graph_for_example_adjacency_matrix.svg.png" decoding="async" width="701" height="217" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Graph_for_example_adjacency_matrix.svg/1052px-Graph_for_example_adjacency_matrix.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d1/Graph_for_example_adjacency_matrix.svg/1402px-Graph_for_example_adjacency_matrix.svg.png 2x" data-file-width="701" data-file-height="217" /></a></span> </p><p>2. Any matrix of the form </p> <dl><dd><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A=\left[{\begin{array}{ccccc}&\vdots &&\vdots \\\dotsb &+1&\dotsb &+1&\dotsb \\&\vdots &&\vdots \\\dotsb &+1&\dotsb &-1&\dotsb \\&\vdots &&\vdots \end{array}}\right].}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="center center center center center" rowspacing="4pt" columnspacing="1em"> <mtr> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> <mtr> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo>+</mo> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> <mtd> <mo>−</mo> <mn>1</mn> </mtd> <mtd> <mo>⋯</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mo>⋮</mo> </mtd> <mtd /> <mtd> <mo>⋮</mo> </mtd> </mtr> </mtable> </mrow> <mo>]</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A=\left[{\begin{array}{ccccc}&\vdots &&\vdots \\\dotsb &+1&\dotsb &+1&\dotsb \\&\vdots &&\vdots \\\dotsb &+1&\dotsb &-1&\dotsb \\&\vdots &&\vdots \end{array}}\right].}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5f014603aec5e966151e4c0234d13a58c585f2e4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.005ex; width:33.129ex; height:19.176ex;" alt="{\displaystyle A=\left[{\begin{array}{ccccc}&\vdots &&\vdots \\\dotsb &+1&\dotsb &+1&\dotsb \\&\vdots &&\vdots \\\dotsb &+1&\dotsb &-1&\dotsb \\&\vdots &&\vdots \end{array}}\right].}"></span></dd></dl> <p>is <i>not</i> totally unimodular, since it has a square submatrix of determinant −2. </p> <div class="mw-heading mw-heading2"><h2 id="Abstract_linear_algebra">Abstract linear algebra</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unimodular_matrix&action=edit&section=5" title="Edit section: Abstract linear algebra"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p><a href="/wiki/Abstract_algebra" title="Abstract algebra">Abstract linear algebra</a> considers matrices with entries from any <a href="/wiki/Commutative" class="mw-redirect" title="Commutative">commutative</a> <a href="/wiki/Ring_(mathematics)" title="Ring (mathematics)">ring</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4b0bfb3769bf24d80e15374dc37b0441e2616e33" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}"></span>, not limited to the integers. In this context, a unimodular matrix is one that is invertible over the ring; equivalently, whose determinant is a <a href="/wiki/Unit_(ring_theory)" title="Unit (ring theory)">unit</a>. This <a href="/wiki/Group_(mathematics)" title="Group (mathematics)">group</a> is denoted <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {GL} _{n}(R)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>GL</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>⁡</mo> <mo stretchy="false">(</mo> <mi>R</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \operatorname {GL} _{n}(R)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9e51c134d4223af9993e7785a493796b37ff7c86" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.069ex; height:2.843ex;" alt="{\displaystyle \operatorname {GL} _{n}(R)}"></span>.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> A rectangular <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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}"></span>-by-<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}"></span> matrix is said to be unimodular if it can be extended with <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m-k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> <mo>−</mo> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m-k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2fd37199830688e5a44d9be1ad756636b2ba45" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.092ex; height:2.343ex;" alt="{\displaystyle m-k}"></span> rows in <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R^{m}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>m</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle R^{m}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a08a239e7a2966db4af2e766388d97cd839fd831" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.439ex; height:2.343ex;" alt="{\displaystyle R^{m}}"></span> to a unimodular square matrix.<sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> </p><p>Over a <a href="/wiki/Field_(mathematics)" title="Field (mathematics)">field</a>, <i>unimodular</i> has the same meaning as <i><a href="/wiki/Invertible_matrix" title="Invertible matrix">non-singular</a></i>. <i>Unimodular</i> here refers to matrices with coefficients in some ring (often the integers) which are invertible over that ring, and one uses <i>non-singular</i> to mean matrices that are invertible over the field. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unimodular_matrix&action=edit&section=6" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Balanced_matrix" title="Balanced matrix">Balanced matrix</a></li> <li><a href="/wiki/Regular_matroid" title="Regular matroid">Regular matroid</a></li> <li><a href="/wiki/Special_linear_group" title="Special linear group">Special linear group</a></li> <li><a href="/wiki/Total_dual_integrality" title="Total dual integrality">Total dual integrality</a></li> <li><a href="/wiki/Hermite_normal_form" title="Hermite normal form">Hermite normal form</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unimodular_matrix&action=edit&section=7" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">The term was coined by <a href="/wiki/Claude_Berge" title="Claude Berge">Claude Berge</a>, see <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 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Springer. p. 510, Section XIII.3. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-387-95385-X" title="Special:BookSources/0-387-95385-X"><bdi>0-387-95385-X</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Algebra&rft.pages=510%2C+Section+XIII.3&rft.edition=rev.+3rd&rft.pub=Springer&rft.date=2002&rft.isbn=0-387-95385-X&rft.aulast=Lang&rft.aufirst=Serge&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnimodular+matrix" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRosenthalMazeWagner2011" class="citation cs2">Rosenthal, J.; Maze, G.; Wagner, U. (2011), <i>Natural Density of Rectangular Unimodular Integer Matrices</i>, Linear Algebra and its applications, vol. 434, Elsevier, pp. 1319–1324</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Natural+Density+of+Rectangular+Unimodular+Integer+Matrices&rft.series=Linear+Algebra+and+its+applications&rft.pages=1319-1324&rft.pub=Elsevier&rft.date=2011&rft.aulast=Rosenthal&rft.aufirst=J.&rft.au=Maze%2C+G.&rft.au=Wagner%2C+U.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnimodular+matrix" class="Z3988"></span></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMicheliSchnyder2016" class="citation cs2">Micheli, G.; Schnyder, R. (2016), <i>The density of unimodular matrices over integrally closed subrings of function fields</i>, Contemporary Developments in Finite Fields and Applications, World Scientific, pp. 244–253</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+density+of+unimodular+matrices+over+integrally+closed+subrings+of+function+fields&rft.series=Contemporary+Developments+in+Finite+Fields+and+Applications&rft.pages=244-253&rft.pub=World+Scientific&rft.date=2016&rft.aulast=Micheli&rft.aufirst=G.&rft.au=Schnyder%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnimodular+matrix" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGuoYang2013" class="citation cs2">Guo, X.; Yang, G. (2013), <i>The probability of rectangular unimodular matrices over Fq [x]</i>, Linear algebra and its applications, Elsevier, pp. 2675–2682</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+probability+of+rectangular+unimodular+matrices+over+Fq+%5Bx%5D&rft.series=Linear+algebra+and+its+applications&rft.pages=2675-2682&rft.pub=Elsevier&rft.date=2013&rft.aulast=Guo&rft.aufirst=X.&rft.au=Yang%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnimodular+matrix" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unimodular_matrix&action=edit&section=8" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPapadimitriouSteiglitz1998" class="citation cs2">Papadimitriou, Christos H.; Steiglitz, Kenneth (1998), "Section 13.2", <a rel="nofollow" class="external text" href="https://books.google.com/books?id=cDY-joeCGoIC&pg=PA316"><i>Combinatorial Optimization: Algorithms and Complexity</i></a>, Mineola, N.Y.: Dover Publications, p. 316, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-40258-1" title="Special:BookSources/978-0-486-40258-1"><bdi>978-0-486-40258-1</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Section+13.2&rft.btitle=Combinatorial+Optimization%3A+Algorithms+and+Complexity&rft.place=Mineola%2C+N.Y.&rft.pages=316&rft.pub=Dover+Publications&rft.date=1998&rft.isbn=978-0-486-40258-1&rft.aulast=Papadimitriou&rft.aufirst=Christos+H.&rft.au=Steiglitz%2C+Kenneth&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DcDY-joeCGoIC%26pg%3DPA316&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnimodular+matrix" class="Z3988"></span></li> <li><a href="/wiki/Alexander_Schrijver" title="Alexander Schrijver">Alexander Schrijver</a> (1998), <i>Theory of Linear and Integer Programming</i>. John Wiley & Sons, <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-471-98232-6" title="Special:BookSources/0-471-98232-6">0-471-98232-6</a> (mathematical)</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAlexander_Schrijver2003" class="citation cs2"><a href="/wiki/Alexander_Schrijver" title="Alexander Schrijver">Alexander Schrijver</a> (2003), <i>Combinatorial Optimization: Polyhedra and Efficiency</i>, Springer</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Combinatorial+Optimization%3A+Polyhedra+and+Efficiency&rft.pub=Springer&rft.date=2003&rft.au=Alexander+Schrijver&rfr_id=info%3Asid%2Fen.wikipedia.org%3AUnimodular+matrix" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Unimodular_matrix&action=edit&section=9" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://glossary.informs.org/ver2/mpgwiki/index.php?title=Unimodular_matrix">Mathematical Programming Glossary by Harvey J. Greenberg</a></li> <li><a rel="nofollow" class="external text" href="http://mathworld.wolfram.com/UnimodularMatrix.html">Unimodular Matrix from MathWorld</a></li> <li><a rel="nofollow" class="external text" href="http://matthiaswalter.org/TUtest/">Software for testing total unimodularity by M. Walter and K. Truemper</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist 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style="width:1%">Explicitly constrained entries</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alternant_matrix" title="Alternant matrix">Alternant</a></li> <li><a href="/wiki/Anti-diagonal_matrix" title="Anti-diagonal matrix">Anti-diagonal</a></li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Anti-Hermitian</a></li> <li><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Anti-symmetric</a></li> <li><a href="/wiki/Arrowhead_matrix" title="Arrowhead matrix">Arrowhead</a></li> <li><a href="/wiki/Band_matrix" title="Band matrix">Band</a></li> <li><a href="/wiki/Bidiagonal_matrix" title="Bidiagonal matrix">Bidiagonal</a></li> <li><a href="/wiki/Bisymmetric_matrix" title="Bisymmetric matrix">Bisymmetric</a></li> <li><a href="/wiki/Block-diagonal_matrix" class="mw-redirect" title="Block-diagonal matrix">Block-diagonal</a></li> <li><a href="/wiki/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Block_tridiagonal_matrix" class="mw-redirect" title="Block tridiagonal matrix">Block tridiagonal</a></li> <li><a href="/wiki/Boolean_matrix" title="Boolean matrix">Boolean</a></li> <li><a href="/wiki/Cauchy_matrix" title="Cauchy matrix">Cauchy</a></li> <li><a href="/wiki/Centrosymmetric_matrix" title="Centrosymmetric matrix">Centrosymmetric</a></li> <li><a href="/wiki/Conference_matrix" title="Conference matrix">Conference</a></li> <li><a href="/wiki/Complex_Hadamard_matrix" title="Complex Hadamard matrix">Complex Hadamard</a></li> <li><a href="/wiki/Copositive_matrix" title="Copositive matrix">Copositive</a></li> <li><a href="/wiki/Diagonally_dominant_matrix" title="Diagonally dominant matrix">Diagonally dominant</a></li> <li><a href="/wiki/Diagonal_matrix" title="Diagonal matrix">Diagonal</a></li> <li><a href="/wiki/DFT_matrix" title="DFT matrix">Discrete Fourier Transform</a></li> <li><a href="/wiki/Elementary_matrix" title="Elementary matrix">Elementary</a></li> <li><a href="/wiki/Equivalent_matrix" class="mw-redirect" title="Equivalent matrix">Equivalent</a></li> <li><a href="/wiki/Frobenius_matrix" title="Frobenius matrix">Frobenius</a></li> <li><a href="/wiki/Generalized_permutation_matrix" title="Generalized permutation matrix">Generalized permutation</a></li> <li><a href="/wiki/Hadamard_matrix" title="Hadamard matrix">Hadamard</a></li> <li><a href="/wiki/Hankel_matrix" title="Hankel matrix">Hankel</a></li> <li><a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a></li> <li><a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg</a></li> <li><a href="/wiki/Hollow_matrix" title="Hollow matrix">Hollow</a></li> <li><a href="/wiki/Integer_matrix" title="Integer matrix">Integer</a></li> <li><a href="/wiki/Logical_matrix" title="Logical matrix">Logical</a></li> <li><a href="/wiki/Matrix_unit" title="Matrix unit">Matrix unit</a></li> <li><a href="/wiki/Metzler_matrix" title="Metzler matrix">Metzler</a></li> <li><a href="/wiki/Moore_matrix" title="Moore matrix">Moore</a></li> <li><a href="/wiki/Nonnegative_matrix" title="Nonnegative matrix">Nonnegative</a></li> <li><a href="/wiki/Pentadiagonal_matrix" class="mw-redirect" title="Pentadiagonal matrix">Pentadiagonal</a></li> <li><a href="/wiki/Permutation_matrix" title="Permutation matrix">Permutation</a></li> <li><a href="/wiki/Persymmetric_matrix" title="Persymmetric matrix">Persymmetric</a></li> <li><a href="/wiki/Polynomial_matrix" title="Polynomial matrix">Polynomial</a></li> <li><a href="/wiki/Quaternionic_matrix" title="Quaternionic matrix">Quaternionic</a></li> <li><a href="/wiki/Signature_matrix" title="Signature matrix">Signature</a></li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian</a></li> <li><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Skew-symmetric</a></li> <li><a href="/wiki/Skyline_matrix" title="Skyline matrix">Skyline</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse</a></li> <li><a href="/wiki/Sylvester_matrix" title="Sylvester matrix">Sylvester</a></li> <li><a href="/wiki/Symmetric_matrix" title="Symmetric matrix">Symmetric</a></li> <li><a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz</a></li> <li><a href="/wiki/Triangular_matrix" title="Triangular matrix">Triangular</a></li> <li><a href="/wiki/Tridiagonal_matrix" title="Tridiagonal matrix">Tridiagonal</a></li> <li><a href="/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde</a></li> <li><a href="/wiki/Walsh_matrix" title="Walsh matrix">Walsh</a></li> <li><a href="/wiki/Z-matrix_(mathematics)" title="Z-matrix (mathematics)">Z</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Constant</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Exchange_matrix" title="Exchange matrix">Exchange</a></li> <li><a href="/wiki/Hilbert_matrix" title="Hilbert matrix">Hilbert</a></li> <li><a href="/wiki/Identity_matrix" title="Identity matrix">Identity</a></li> <li><a href="/wiki/Lehmer_matrix" title="Lehmer matrix">Lehmer</a></li> <li><a href="/wiki/Matrix_of_ones" title="Matrix of ones">Of ones</a></li> <li><a href="/wiki/Pascal_matrix" title="Pascal matrix">Pascal</a></li> <li><a href="/wiki/Pauli_matrices" title="Pauli matrices">Pauli</a></li> <li><a href="/wiki/Redheffer_matrix" title="Redheffer matrix">Redheffer</a></li> <li><a href="/wiki/Shift_matrix" title="Shift matrix">Shift</a></li> <li><a href="/wiki/Zero_matrix" title="Zero matrix">Zero</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Conditions on <a href="/wiki/Eigenvalues_and_eigenvectors" title="Eigenvalues and eigenvectors">eigenvalues or eigenvectors</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Companion_matrix" title="Companion matrix">Companion</a></li> <li><a href="/wiki/Convergent_matrix" title="Convergent matrix">Convergent</a></li> <li><a href="/wiki/Defective_matrix" title="Defective matrix">Defective</a></li> <li><a href="/wiki/Definite_matrix" title="Definite matrix">Definite</a></li> <li><a href="/wiki/Diagonalizable_matrix" title="Diagonalizable matrix">Diagonalizable</a></li> <li><a href="/wiki/Hurwitz-stable_matrix" title="Hurwitz-stable matrix">Hurwitz-stable</a></li> <li><a href="/wiki/Positive-definite_matrix" class="mw-redirect" title="Positive-definite matrix">Positive-definite</a></li> <li><a href="/wiki/Stieltjes_matrix" title="Stieltjes matrix">Stieltjes</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satisfying conditions on <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">products</a> or <a href="/wiki/Inverse_of_a_matrix" class="mw-redirect" title="Inverse of a matrix">inverses</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Matrix_congruence" title="Matrix congruence">Congruent</a></li> <li><a href="/wiki/Idempotent_matrix" title="Idempotent matrix">Idempotent</a> or <a href="/wiki/Projection_(linear_algebra)" title="Projection (linear algebra)">Projection</a></li> <li><a href="/wiki/Invertible_matrix" title="Invertible matrix">Invertible</a></li> <li><a href="/wiki/Involutory_matrix" title="Involutory matrix">Involutory</a></li> <li><a href="/wiki/Nilpotent_matrix" title="Nilpotent matrix">Nilpotent</a></li> <li><a href="/wiki/Normal_matrix" title="Normal matrix">Normal</a></li> <li><a href="/wiki/Orthogonal_matrix" title="Orthogonal matrix">Orthogonal</a></li> <li><a class="mw-selflink selflink">Unimodular</a></li> <li><a href="/wiki/Unipotent" title="Unipotent">Unipotent</a></li> <li><a href="/wiki/Unitary_matrix" title="Unitary matrix">Unitary</a></li> <li><a href="/wiki/Totally_unimodular_matrix" class="mw-redirect" title="Totally unimodular matrix">Totally unimodular</a></li> <li><a href="/wiki/Weighing_matrix" title="Weighing matrix">Weighing</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">With specific applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjugate_matrix" title="Adjugate matrix">Adjugate</a></li> <li><a href="/wiki/Alternating_sign_matrix" title="Alternating sign matrix">Alternating sign</a></li> <li><a href="/wiki/Augmented_matrix" title="Augmented matrix">Augmented</a></li> <li><a href="/wiki/B%C3%A9zout_matrix" title="Bézout matrix">Bézout</a></li> <li><a href="/wiki/Carleman_matrix" title="Carleman matrix">Carleman</a></li> <li><a href="/wiki/Cartan_matrix" title="Cartan matrix">Cartan</a></li> <li><a href="/wiki/Circulant_matrix" title="Circulant matrix">Circulant</a></li> <li><a href="/wiki/Cofactor_matrix" class="mw-redirect" title="Cofactor matrix">Cofactor</a></li> <li><a href="/wiki/Commutation_matrix" title="Commutation matrix">Commutation</a></li> <li><a href="/wiki/Confusion_matrix" title="Confusion matrix">Confusion</a></li> <li><a href="/wiki/Coxeter_matrix" class="mw-redirect" title="Coxeter matrix">Coxeter</a></li> <li><a href="/wiki/Distance_matrix" title="Distance matrix">Distance</a></li> <li><a href="/wiki/Duplication_and_elimination_matrices" title="Duplication and elimination matrices">Duplication and elimination</a></li> <li><a href="/wiki/Euclidean_distance_matrix" title="Euclidean distance matrix">Euclidean distance</a></li> <li><a href="/wiki/Fundamental_matrix_(linear_differential_equation)" title="Fundamental matrix (linear differential equation)">Fundamental (linear differential equation)</a></li> <li><a href="/wiki/Generator_matrix" title="Generator matrix">Generator</a></li> <li><a href="/wiki/Gram_matrix" title="Gram matrix">Gram</a></li> <li><a href="/wiki/Hessian_matrix" title="Hessian matrix">Hessian</a></li> <li><a href="/wiki/Householder_transformation" title="Householder transformation">Householder</a></li> <li><a href="/wiki/Jacobian_matrix_and_determinant" title="Jacobian matrix and determinant">Jacobian</a></li> <li><a href="/wiki/Moment_matrix" title="Moment matrix">Moment</a></li> <li><a href="/wiki/Payoff_matrix" class="mw-redirect" title="Payoff matrix">Payoff</a></li> <li><a href="/wiki/Pick_matrix" class="mw-redirect" title="Pick matrix">Pick</a></li> <li><a href="/wiki/Random_matrix" title="Random matrix">Random</a></li> <li><a href="/wiki/Rotation_matrix" title="Rotation matrix">Rotation</a></li> <li><a href="/wiki/Routh%E2%80%93Hurwitz_matrix" title="Routh–Hurwitz matrix">Routh-Hurwitz</a></li> <li><a href="/wiki/Seifert_matrix" class="mw-redirect" title="Seifert matrix">Seifert</a></li> <li><a href="/wiki/Shear_matrix" class="mw-redirect" title="Shear matrix">Shear</a></li> <li><a href="/wiki/Similarity_matrix" class="mw-redirect" title="Similarity matrix">Similarity</a></li> <li><a href="/wiki/Symplectic_matrix" title="Symplectic matrix">Symplectic</a></li> <li><a href="/wiki/Totally_positive_matrix" title="Totally positive matrix">Totally positive</a></li> <li><a href="/wiki/Transformation_matrix" title="Transformation matrix">Transformation</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/wiki/Statistics" title="Statistics">statistics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Centering_matrix" title="Centering matrix">Centering</a></li> <li><a href="/wiki/Correlation_matrix" class="mw-redirect" title="Correlation matrix">Correlation</a></li> <li><a href="/wiki/Covariance_matrix" title="Covariance matrix">Covariance</a></li> <li><a href="/wiki/Design_matrix" title="Design matrix">Design</a></li> <li><a href="/wiki/Doubly_stochastic_matrix" title="Doubly stochastic matrix">Doubly stochastic</a></li> <li><a href="/wiki/Fisher_information_matrix" class="mw-redirect" title="Fisher information matrix">Fisher information</a></li> <li><a href="/wiki/Projection_matrix" title="Projection matrix">Hat</a></li> <li><a href="/wiki/Precision_(statistics)" title="Precision (statistics)">Precision</a></li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Stochastic</a></li> <li><a href="/wiki/Stochastic_matrix" title="Stochastic matrix">Transition</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Adjacency_matrix" title="Adjacency matrix">Adjacency</a></li> <li><a href="/wiki/Biadjacency_matrix" class="mw-redirect" title="Biadjacency matrix">Biadjacency</a></li> <li><a href="/wiki/Degree_matrix" title="Degree matrix">Degree</a></li> <li><a href="/wiki/Edmonds_matrix" title="Edmonds matrix">Edmonds</a></li> <li><a href="/wiki/Incidence_matrix" title="Incidence matrix">Incidence</a></li> <li><a href="/wiki/Laplacian_matrix" title="Laplacian matrix">Laplacian</a></li> <li><a href="/wiki/Seidel_adjacency_matrix" title="Seidel adjacency matrix">Seidel adjacency</a></li> <li><a href="/wiki/Tutte_matrix" title="Tutte matrix">Tutte</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Used in science and engineering</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Cabibbo%E2%80%93Kobayashi%E2%80%93Maskawa_matrix" title="Cabibbo–Kobayashi–Maskawa matrix">Cabibbo–Kobayashi–Maskawa</a></li> <li><a href="/wiki/Density_matrix" title="Density matrix">Density</a></li> <li><a href="/wiki/Fundamental_matrix_(computer_vision)" title="Fundamental matrix (computer vision)">Fundamental (computer vision)</a></li> <li><a href="/wiki/Fuzzy_associative_matrix" title="Fuzzy associative matrix">Fuzzy associative</a></li> <li><a href="/wiki/Gamma_matrices" title="Gamma matrices">Gamma</a></li> <li><a href="/wiki/Gell-Mann_matrices" title="Gell-Mann matrices">Gell-Mann</a></li> <li><a href="/wiki/Hamiltonian_matrix" title="Hamiltonian matrix">Hamiltonian</a></li> <li><a href="/wiki/Irregular_matrix" title="Irregular matrix">Irregular</a></li> <li><a href="/wiki/Overlap_matrix" class="mw-redirect" title="Overlap matrix">Overlap</a></li> <li><a href="/wiki/S-matrix" title="S-matrix">S</a></li> <li><a href="/wiki/State-transition_matrix" title="State-transition matrix">State transition</a></li> <li><a href="/wiki/Substitution_matrix" title="Substitution matrix">Substitution</a></li> <li><a href="/wiki/Z-matrix_(chemistry)" title="Z-matrix (chemistry)">Z (chemistry)</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related terms</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jordan_normal_form" title="Jordan normal form">Jordan normal form</a></li> <li><a href="/wiki/Linear_independence" title="Linear independence">Linear independence</a></li> <li><a href="/wiki/Matrix_exponential" title="Matrix exponential">Matrix exponential</a></li> <li><a href="/wiki/Matrix_representation_of_conic_sections" title="Matrix representation of conic sections">Matrix representation of conic sections</a></li> <li><a href="/wiki/Perfect_matrix" title="Perfect matrix">Perfect matrix</a></li> <li><a href="/wiki/Pseudoinverse" class="mw-redirect" title="Pseudoinverse">Pseudoinverse</a></li> <li><a href="/wiki/Row_echelon_form" title="Row echelon form">Row echelon form</a></li> <li><a href="/wiki/Wronskian" title="Wronskian">Wronskian</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div> <ul><li><b><span class="nowrap"><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Nuvola_apps_edu_mathematics_blue-p.svg" class="mw-file-description"><img alt="icon" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/16px-Nuvola_apps_edu_mathematics_blue-p.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a 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