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mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%94%CF%85%CE%B1%CE%B4%CE%B9%CE%BA%CF%8C%CF%82_%CF%80%CE%AF%CE%BD%CE%B1%CE%BA%CE%B1%CF%82" title="Δυαδικός πίνακας – Greek" lang="el" hreflang="el" data-title="Δυαδικός πίνακας" 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_booleana" title="Matriz booleana – Spanish" lang="es" hreflang="es" data-title="Matriz booleana" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D8%A7%D8%AA%D8%B1%DB%8C%D8%B3_%D9%85%D9%86%D8%B7%D9%82%DB%8C" title="ماتریس منطقی – Persian" lang="fa" hreflang="fa" data-title="ماتریس منطقی" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Matrice_binaire" title="Matrice binaire – French" lang="fr" hreflang="fr" data-title="Matrice binaire" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EC%9D%B4%EC%A7%84_%ED%96%89%EB%A0%AC" title="이진 행렬 – Korean" lang="ko" hreflang="ko" data-title="이진 행렬" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-it mw-list-item"><a href="https://it.wikipedia.org/wiki/Matrice_binaria" title="Matrice binaria – Italian" lang="it" hreflang="it" data-title="Matrice binaria" data-language-autonym="Italiano" data-language-local-name="Italian" class="interlanguage-link-target"><span>Italiano</span></a></li><li class="interlanguage-link interwiki-kk mw-list-item"><a href="https://kk.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D1%8C_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0%D1%81%D1%8B" title="Буль матрицасы – Kazakh" lang="kk" hreflang="kk" data-title="Буль матрицасы" data-language-autonym="Қазақша" data-language-local-name="Kazakh" class="interlanguage-link-target"><span>Қазақша</span></a></li><li class="interlanguage-link interwiki-nl mw-list-item"><a href="https://nl.wikipedia.org/wiki/Binaire_matrix" title="Binaire matrix – Dutch" lang="nl" hreflang="nl" data-title="Binaire matrix" data-language-autonym="Nederlands" data-language-local-name="Dutch" class="interlanguage-link-target"><span>Nederlands</span></a></li><li class="interlanguage-link interwiki-pl mw-list-item"><a href="https://pl.wikipedia.org/wiki/Macierz_logiczna" title="Macierz logiczna – Polish" lang="pl" hreflang="pl" data-title="Macierz logiczna" data-language-autonym="Polski" data-language-local-name="Polish" class="interlanguage-link-target"><span>Polski</span></a></li><li class="interlanguage-link interwiki-pt mw-list-item"><a href="https://pt.wikipedia.org/wiki/Matriz_booliana" title="Matriz booliana – Portuguese" lang="pt" hreflang="pt" data-title="Matriz booliana" data-language-autonym="Português" data-language-local-name="Portuguese" class="interlanguage-link-target"><span>Português</span></a></li><li class="interlanguage-link interwiki-ru mw-list-item"><a href="https://ru.wikipedia.org/wiki/%D0%91%D0%B8%D0%BD%D0%B0%D1%80%D0%BD%D0%B0%D1%8F_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Бинарная матрица – Russian" lang="ru" hreflang="ru" data-title="Бинарная матрица" data-language-autonym="Русский" data-language-local-name="Russian" class="interlanguage-link-target"><span>Русский</span></a></li><li class="interlanguage-link interwiki-sl mw-list-item"><a href="https://sl.wikipedia.org/wiki/Binarna_matrika" title="Binarna matrika – Slovenian" lang="sl" hreflang="sl" data-title="Binarna matrika" data-language-autonym="Slovenščina" data-language-local-name="Slovenian" class="interlanguage-link-target"><span>Slovenščina</span></a></li><li class="interlanguage-link interwiki-sr mw-list-item"><a href="https://sr.wikipedia.org/wiki/%D0%91%D1%83%D0%BB%D0%BE%D0%B2%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D0%B0" title="Булова матрица – Serbian" lang="sr" hreflang="sr" data-title="Булова матрица" data-language-autonym="Српски / srpski" data-language-local-name="Serbian" class="interlanguage-link-target"><span>Српски / srpski</span></a></li><li class="interlanguage-link interwiki-sv mw-list-item"><a href="https://sv.wikipedia.org/wiki/Bin%C3%A4r_matris" title="Binär matris – Swedish" lang="sv" hreflang="sv" data-title="Binär matris" data-language-autonym="Svenska" data-language-local-name="Swedish" class="interlanguage-link-target"><span>Svenska</span></a></li><li class="interlanguage-link interwiki-ta mw-list-item"><a href="https://ta.wikipedia.org/wiki/%E0%AE%A4%E0%AE%B0%E0%AF%81%E0%AE%95%E0%AF%8D%E0%AE%95_%E0%AE%85%E0%AE%A3%E0%AE%BF" title="தருக்க அணி – Tamil" lang="ta" hreflang="ta" data-title="தருக்க அணி" data-language-autonym="தமிழ்" data-language-local-name="Tamil" class="interlanguage-link-target"><span>தமிழ்</span></a></li><li class="interlanguage-link interwiki-th mw-list-item"><a href="https://th.wikipedia.org/wiki/%E0%B9%80%E0%B8%A1%E0%B8%97%E0%B8%A3%E0%B8%B4%E0%B8%81%E0%B8%8B%E0%B9%8C%E0%B9%80%E0%B8%8A%E0%B8%B4%E0%B8%87%E0%B8%95%E0%B8%A3%E0%B8%A3%E0%B8%81%E0%B8%B0" title="เมทริกซ์เชิงตรรกะ – Thai" lang="th" hreflang="th" data-title="เมทริกซ์เชิงตรรกะ" data-language-autonym="ไทย" data-language-local-name="Thai" class="interlanguage-link-target"><span>ไทย</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%91%D1%96%D0%BD%D0%B0%D1%80%D0%BD%D0%B0_%D0%BC%D0%B0%D1%82%D1%80%D0%B8%D1%86%D1%8F" title="Бінарна матриця – Ukrainian" lang="uk" hreflang="uk" data-title="Бінарна матриця" data-language-autonym="Українська" data-language-local-name="Ukrainian" class="interlanguage-link-target"><span>Українська</span></a></li><li class="interlanguage-link interwiki-zh-yue mw-list-item"><a href="https://zh-yue.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E7%9F%A9%E9%99%A3" title="邏輯矩陣 – Cantonese" lang="yue" hreflang="yue" data-title="邏輯矩陣" data-language-autonym="粵語" data-language-local-name="Cantonese" class="interlanguage-link-target"><span>粵語</span></a></li><li class="interlanguage-link interwiki-zh mw-list-item"><a href="https://zh.wikipedia.org/wiki/%E9%82%8F%E8%BC%AF%E7%9F%A9%E9%99%A3" title="邏輯矩陣 – Chinese" lang="zh" hreflang="zh" data-title="邏輯矩陣" data-language-autonym="中文" data-language-local-name="Chinese" 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id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Matrix of binary truth values</div> <p>A <b>logical matrix</b>, <b>binary matrix</b>, <b>relation matrix</b>, <b>Boolean matrix</b>, or <b>(0, 1)-matrix</b> is a <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> with entries from the <a href="/wiki/Boolean_domain" title="Boolean domain">Boolean domain</a> <span class="texhtml"><b>B</b> = {0, 1}.</span> Such a matrix can be used to represent a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> between a pair of <a href="/wiki/Finite_set" title="Finite set">finite sets</a>. It is an important tool in <a href="/wiki/Combinatorial_mathematics" class="mw-redirect" title="Combinatorial mathematics">combinatorial mathematics</a> and <a href="/wiki/Theoretical_computer_science" title="Theoretical computer science">theoretical computer science</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Matrix_representation_of_a_relation">Matrix representation of a relation</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_matrix&action=edit&section=1" title="Edit section: Matrix representation of a relation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>If <i>R</i> is a <a href="/wiki/Binary_relation" title="Binary relation">binary relation</a> between the finite <a href="/wiki/Indexed_set" class="mw-redirect" title="Indexed set">indexed sets</a> <i>X</i> and <i>Y</i> (so <span class="texhtml"> <i>R</i> ⊆ <i>X</i> ×<i>Y</i></span>), then <i>R</i> can be represented by the logical matrix <i>M</i> whose row and column indices index the elements of <i>X</i> and <i>Y</i>, respectively, such that the entries of <i>M</i> are defined by </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 m_{i,j}={\begin{cases}1&(x_{i},y_{j})\in R,\\0&(x_{i},y_{j})\not \in R.\end{cases}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mo>,</mo> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>{</mo> <mtable columnalign="left left" rowspacing=".2em" columnspacing="1em" displaystyle="false"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∈</mo> <mi>R</mi> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mi>y</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>∉</mo> <mi>R</mi> <mo>.</mo> </mtd> </mtr> </mtable> <mo fence="true" stretchy="true" symmetric="true"></mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{i,j}={\begin{cases}1&(x_{i},y_{j})\in R,\\0&(x_{i},y_{j})\not \in R.\end{cases}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/41ca05acfcf752dc2c06f4c16e42c69c30885a14" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.326ex; height:6.176ex;" alt="{\displaystyle m_{i,j}={\begin{cases}1&(x_{i},y_{j})\in R,\\0&(x_{i},y_{j})\not \in R.\end{cases}}}"></span></dd></dl> <p>In order to designate the row and column numbers of the matrix, the sets <i>X</i> and <i>Y</i> are indexed with positive <a href="/wiki/Integer" title="Integer">integers</a>: <i>i</i> ranges from 1 to the <a href="/wiki/Cardinality" title="Cardinality">cardinality</a> (size) of <i>X</i>, and <i>j</i> ranges from 1 to the cardinality of <i>Y</i>. See the article on <a href="/wiki/Indexed_set" class="mw-redirect" title="Indexed set">indexed sets</a> for more detail. </p> <div class="mw-heading mw-heading3"><h3 id="Example">Example</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_matrix&action=edit&section=2" title="Edit section: Example"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The binary relation <i>R</i> on the set <span class="texhtml">{1, 2, 3, 4}</span> is defined so that <i>aRb</i> holds if and only if <i>a</i> <a href="/wiki/Divides" class="mw-redirect" title="Divides">divides</a> <i>b</i> evenly, with no remainder. For example, 2<i>R</i>4 holds because 2 divides 4 without leaving a remainder, but 3<i>R</i>4 does not hold because when 3 divides 4, there is a remainder of 1. The following set is the set of pairs for which the relation <i>R</i> holds. </p> <dl><dd>{(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}.</dd></dl> <p>The corresponding representation as a logical matrix is </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 {\begin{pmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{pmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>(</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>0</mn> </mtd> </mtr> <mtr> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>0</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>)</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{pmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{pmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a94f56e692dc674073b8b46436b13d96c0b8b7b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:17.083ex; height:12.509ex;" alt="{\displaystyle {\begin{pmatrix}1&1&1&1\\0&1&0&1\\0&0&1&0\\0&0&0&1\end{pmatrix}},}"></span></dd></dl> <p>which includes a diagonal of ones, since each number divides itself. </p> <div class="mw-heading mw-heading2"><h2 id="Other_examples">Other examples</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_matrix&action=edit&section=3" title="Edit section: Other examples"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>A <a href="/wiki/Permutation_matrix" title="Permutation matrix">permutation matrix</a> is a (0, 1)-matrix, all of whose columns and rows each have exactly one nonzero element. <ul><li>A <a href="/wiki/Costas_array" title="Costas array">Costas array</a> is a special case of a permutation matrix.</li></ul></li> <li>An <a href="/wiki/Incidence_matrix" title="Incidence matrix">incidence matrix</a> in <a href="/wiki/Combinatorics" title="Combinatorics">combinatorics</a> and <a href="/wiki/Finite_geometry" title="Finite geometry">finite geometry</a> has ones to indicate incidence between points (or vertices) and lines of a geometry, blocks of a <a href="/wiki/Block_design" title="Block design">block design</a>, or edges of a <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graph</a>.</li> <li>A <a href="/wiki/Design_matrix" title="Design matrix">design matrix</a> in <a href="/wiki/Analysis_of_variance" title="Analysis of variance">analysis of variance</a> is a (0, 1)-matrix with constant row sums.</li> <li>A logical matrix may represent an <a href="/wiki/Adjacency_matrix" title="Adjacency matrix">adjacency matrix</a> in <a href="/wiki/Graph_theory" title="Graph theory">graph theory</a>: non-<a href="/wiki/Symmetric_matrix" title="Symmetric matrix">symmetric</a> matrices correspond to <a href="/wiki/Directed_graph" title="Directed graph">directed graphs</a>, symmetric matrices to ordinary <a href="/wiki/Graph_(discrete_mathematics)" title="Graph (discrete mathematics)">graphs</a>, and a 1 on the diagonal corresponds to a <a href="/wiki/Loop_(graph_theory)" title="Loop (graph theory)">loop</a> at the corresponding vertex.</li> <li>The <a href="/wiki/Biadjacency_matrix" class="mw-redirect" title="Biadjacency matrix">biadjacency matrix</a> of a simple, undirected <a href="/wiki/Bipartite_graph" title="Bipartite graph">bipartite graph</a> is a (0, 1)-matrix, and any (0, 1)-matrix arises in this way.</li> <li>The <a href="/wiki/Prime_factor" class="mw-redirect" title="Prime factor">prime factors</a> of a list of <i>m</i> <a href="/wiki/Square-free_integer" title="Square-free integer">square-free</a>, <a href="/wiki/Smooth_number" title="Smooth number"><i>n</i>-smooth</a> numbers can be described as an <i>m</i> × π(<i>n</i>) (0, 1)-matrix, where π is the <a href="/wiki/Prime-counting_function" title="Prime-counting function">prime-counting function</a>, and <i>a</i><sub><i>ij</i></sub> is 1 if and only if the <i>j</i>th prime divides the <i>i</i>th number. This representation is useful in the <a href="/wiki/Quadratic_sieve" title="Quadratic sieve">quadratic sieve</a> factoring algorithm.</li> <li>A <a href="/wiki/Raster_graphics" title="Raster graphics">bitmap image</a> containing <a href="/wiki/Pixel" title="Pixel">pixels</a> in only two colors can be represented as a (0, 1)-matrix in which the zeros represent pixels of one color and the ones represent pixels of the other color.</li> <li>A binary matrix can be used to check the game rules in the game of <a href="/wiki/Go_(game)" title="Go (game)">Go</a>.<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></li> <li>The <a href="/wiki/Four-valued_logic#Matrix_transitions" title="Four-valued logic">four valued logic</a> of two bits, transformed by 2x2 logical matrices, forms a <a href="/wiki/Transition_system" title="Transition system">transition system</a>.</li> <li>A <a href="/wiki/Recurrence_plot" title="Recurrence plot">recurrence plot</a> and its variants are matrices that shows which pairs of points are closer than a certain vicinity threshold in a <a href="/wiki/Phase_space" title="Phase space">phase space</a>.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Some_properties">Some properties</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_matrix&action=edit&section=4" title="Edit section: Some properties"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Matrix_multiply.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Matrix_multiply.png/220px-Matrix_multiply.png" decoding="async" width="220" height="391" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Matrix_multiply.png/330px-Matrix_multiply.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Matrix_multiply.png/440px-Matrix_multiply.png 2x" data-file-width="600" data-file-height="1065" /></a><figcaption>Multiplication of two logical matrices using <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a>.</figcaption></figure> <p>The matrix representation of the <a href="/wiki/Equality_(mathematics)" title="Equality (mathematics)">equality relation</a> on a finite set is the <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a> <i>I</i>, that is, the matrix whose entries on the diagonal are all 1, while the others are all 0. More generally, if relation <i>R</i> satisfies <span class="texhtml">I ⊆ <i>R</i>,</span> then <i>R</i> is a <a href="/wiki/Reflexive_relation" title="Reflexive relation">reflexive relation</a>. </p><p>If the Boolean domain is viewed as a <a href="/wiki/Semiring" title="Semiring">semiring</a>, where addition corresponds to <a href="/wiki/Logical_OR" class="mw-redirect" title="Logical OR">logical OR</a> and multiplication to <a href="/wiki/Logical_AND" class="mw-redirect" title="Logical AND">logical AND</a>, the matrix representation of the <a href="/wiki/Composition_of_relations" title="Composition of relations">composition</a> of two relations is equal to the <a href="/wiki/Matrix_product" class="mw-redirect" title="Matrix product">matrix product</a> of the matrix representations of these relations. This product can be computed in <a href="/wiki/Expected_value" title="Expected value">expected</a> time O(<i>n</i><sup>2</sup>).<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> </p><p>Frequently, operations on binary matrices are defined in terms of <a href="/wiki/Modular_arithmetic" title="Modular arithmetic">modular arithmetic</a> mod 2—that is, the elements are treated as elements of the <a href="/wiki/Galois_field" class="mw-redirect" title="Galois field">Galois field</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 {\mathbf {GF}}(2)=\mathbb {Z} _{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">G</mi> <mi mathvariant="bold">F</mi> </mrow> </mrow> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">Z</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathbf {GF}}(2)=\mathbb {Z} _{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e702f9fe14d00165f436bc1b2694108ae2c76af9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.458ex; height:2.843ex;" alt="{\displaystyle {\mathbf {GF}}(2)=\mathbb {Z} _{2}}"></span>. They arise in a variety of representations and have a number of more restricted special forms. They are applied e.g. in <a href="/wiki/XOR-satisfiability" class="mw-redirect" title="XOR-satisfiability">XOR-satisfiability</a>. </p><p>The number of distinct <i>m</i>-by-<i>n</i> binary matrices is equal to 2<sup><i>mn</i></sup>, and is thus finite. </p> <div class="mw-heading mw-heading2"><h2 id="Lattice">Lattice</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_matrix&action=edit&section=5" title="Edit section: Lattice"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Let <i>n</i> and <i>m</i> be given and let <i>U</i> denote the set of all logical <i>m</i> × <i>n</i> matrices. Then <i>U</i> has a <a href="/wiki/Partial_order" class="mw-redirect" title="Partial order">partial order</a> given by </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 \forall A,B\in U,\quad A\leq B\quad {\text{when}}\quad \forall i,j\quad A_{ij}=1\implies B_{ij}=1.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∀</mi> <mi>A</mi> <mo>,</mo> <mi>B</mi> <mo>∈</mo> <mi>U</mi> <mo>,</mo> <mspace width="1em" /> <mi>A</mi> <mo>≤</mo> <mi>B</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>when</mtext> </mrow> <mspace width="1em" /> <mi mathvariant="normal">∀</mi> <mi>i</mi> <mo>,</mo> <mi>j</mi> <mspace width="1em" /> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> <mspace width="thickmathspace" /> <mo stretchy="false">⟹</mo> <mspace width="thickmathspace" /> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mn>1.</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \forall A,B\in U,\quad A\leq B\quad {\text{when}}\quad \forall i,j\quad A_{ij}=1\implies B_{ij}=1.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/509a9f78665a7d1f540caf02a5febdf90e274cef" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:58.785ex; height:2.843ex;" alt="{\displaystyle \forall A,B\in U,\quad A\leq B\quad {\text{when}}\quad \forall i,j\quad A_{ij}=1\implies B_{ij}=1.}"></span></dd></dl> <p>In fact, <i>U</i> forms a <a href="/wiki/Boolean_algebra" title="Boolean algebra">Boolean algebra</a> with the operations <a href="/wiki/And_(logic)" class="mw-redirect" title="And (logic)">and</a> & <a href="/wiki/Or_(logic)" class="mw-redirect" title="Or (logic)">or</a> between two matrices applied component-wise. The complement of a logical matrix is obtained by swapping all zeros and ones for their opposite. </p><p>Every logical matrix <span class="texhtml"><i>A</i> = (<i>A</i><sub><i>ij</i></sub>)</span> has a transpose <span class="texhtml"><i>A</i><sup>T</sup> = (<i>A</i><sub><i>ji</i></sub>).</span> Suppose <i>A</i> is a logical matrix with no columns or rows identically zero. Then the matrix product, using Boolean arithmetic, <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^{\operatorname {T} }A}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> <mi>A</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A^{\operatorname {T} }A}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5e20f715e82efaa5a6a003d473dafd5aaf571a1e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.905ex; height:2.676ex;" alt="{\displaystyle A^{\operatorname {T} }A}"></span> contains the <i>m</i> × <i>m</i> <a href="/wiki/Identity_matrix" title="Identity matrix">identity matrix</a>, and the product <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 AA^{\operatorname {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <msup> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle AA^{\operatorname {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa56784c4ea69edef14db6d00cb3b93f661986a7" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.905ex; height:2.676ex;" alt="{\displaystyle AA^{\operatorname {T} }}"></span> contains the <i>n</i> × <i>n</i> identity. </p><p>As a mathematical structure, the Boolean algebra <i>U</i> forms a <a href="/wiki/Lattice_(order)" title="Lattice (order)">lattice</a> ordered by <a href="/wiki/Inclusion_(logic)" title="Inclusion (logic)">inclusion</a>; additionally it is a multiplicative lattice due to matrix multiplication. </p><p>Every logical matrix in <i>U</i> corresponds to a binary relation. These listed operations on <i>U</i>, and ordering, correspond to a <a href="/wiki/Algebraic_logic#Calculus_of_relations" title="Algebraic logic">calculus of relations</a>, where the matrix multiplication represents <a href="/wiki/Composition_of_relations" title="Composition of relations">composition of relations</a>.<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> <div class="mw-heading mw-heading2"><h2 id="Logical_vectors">Logical vectors</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_matrix&action=edit&section=6" title="Edit section: Logical vectors"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <table class="floatright" style="font-size:90%"> <caption style="background:#cee0f2; padding:0.2em 0.4em 0.2em; text-align:center; font-size:130%; line-height:1.2em; font-weight: bold">Group-like structures </caption> <tbody><tr> <th> </th> <th><a href="/wiki/Total_function" class="mw-redirect" title="Total function">Total</a> </th> <th><a href="/wiki/Associative_property" title="Associative property">Associative</a> </th> <th><a href="/wiki/Identity_element" title="Identity element">Identity</a> </th> <th><a href="/wiki/Quasigroup" title="Quasigroup"><style data-mw-deduplicate="TemplateStyles:r1038841319">'"`UNIQ--templatestyles-00000016-QINU`"'</style><span class="rt-commentedText tooltip tooltip-dotted" title="Here, divisibility refers specifically to the quasigroup axioms">Divisible</span></a> </th> <th><a href="/wiki/Commutative_property" title="Commutative property">Commutative</a> </th></tr> <tr> <th><a href="/wiki/Partial_groupoid" title="Partial groupoid">Partial magma</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Semigroupoid" title="Semigroupoid">Semigroupoid</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Category_(mathematics)" title="Category (mathematics)">Small category</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Groupoid" title="Groupoid">Groupoid</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Groupoid" title="Groupoid">groupoid</a> </th> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Magma_(algebra)" title="Magma (algebra)">Magma</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Magma_(algebra)" title="Magma (algebra)">magma</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Quasigroup" title="Quasigroup">Quasigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Unital_magma" class="mw-redirect" title="Unital magma">Unital magma</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Unital_magma" class="mw-redirect" title="Unital magma">unital magma</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Loop_(algebra)" class="mw-redirect" title="Loop (algebra)">Loop</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Loop_(algebra)" class="mw-redirect" title="Loop (algebra)">loop</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Semigroup" title="Semigroup">Semigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative <a href="/wiki/Semigroup" title="Semigroup">semigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th>Associative <a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th>Commutative-and-associative <a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Monoid" title="Monoid">Monoid</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Commutative_monoid" class="mw-redirect" title="Commutative monoid">Commutative monoid</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr> <tr> <th><a href="/wiki/Group_(mathematics)" title="Group (mathematics)">Group</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#FFC7C7;color:black;vertical-align:middle;text-align:center;" class="table-no">Unneeded </td></tr> <tr> <th><a href="/wiki/Abelian_group" title="Abelian group">Abelian group</a> </th> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required</td> <td style="background:#9EFF9E;color:black;vertical-align:middle;text-align:center;" class="table-yes">Required </td></tr></tbody></table> <p>If <i>m</i> or <i>n</i> equals one, then the <i>m</i> × <i>n</i> logical matrix (<i>m</i><sub><i>ij</i></sub>) is a <b>logical vector</b> or <a href="/wiki/Bit_string" class="mw-redirect" title="Bit string">bit string</a>. If <i>m</i> = 1, the vector is a row vector, and if <i>n</i> = 1, it is a column vector. In either case the index equaling 1 is dropped from denotation of the vector. </p><p>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 (P_{i}),\,i=1,2,\ldots ,m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mi>i</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (P_{i}),\,i=1,2,\ldots ,m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05b216556283ee364721493aa3eb7f22794aba66" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.001ex; height:2.843ex;" alt="{\displaystyle (P_{i}),\,i=1,2,\ldots ,m}"></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 (Q_{j}),\,j=1,2,\ldots ,n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo stretchy="false">(</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>,</mo> <mspace width="thinmathspace" /> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mo>…</mo> <mo>,</mo> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle (Q_{j}),\,j=1,2,\ldots ,n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a516bf4de94eb994f025d81bbd6289b13647fcb1" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.967ex; height:3.009ex;" alt="{\displaystyle (Q_{j}),\,j=1,2,\ldots ,n}"></span> are two logical vectors. The <a href="/wiki/Outer_product" title="Outer product">outer product</a> of <i>P</i> and <i>Q</i> results in an <i>m</i> × <i>n</i> <a href="/wiki/Rectangular_relation" class="mw-redirect" title="Rectangular relation">rectangular relation</a> </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 m_{ij}=P_{i}\land Q_{j}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>m</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>P</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>∧</mo> <msub> <mi>Q</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m_{ij}=P_{i}\land Q_{j}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9dc8068dddddb038b3e0cd7f87a86349c66c2bd0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:14.885ex; height:2.843ex;" alt="{\displaystyle m_{ij}=P_{i}\land Q_{j}.}"></span></dd></dl> <p>A reordering of the rows and columns of such a matrix can assemble all the ones into a rectangular part of the matrix.<sup id="cite_ref-GS_4-0" class="reference"><a href="#cite_note-GS-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p><p>Let <i>h</i> be the vector of all ones. Then if <i>v</i> is an arbitrary logical vector, the relation <i>R</i> = <i>v h</i><sup>T</sup> has constant rows determined by <i>v</i>. In the <a href="/wiki/Calculus_of_relations" class="mw-redirect" title="Calculus of relations">calculus of relations</a> such an <i>R</i> is called a vector.<sup id="cite_ref-GS_4-1" class="reference"><a href="#cite_note-GS-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> A particular instance is the universal relation <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 hh^{\operatorname {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>h</mi> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle hh^{\operatorname {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4d71f316dfa762a0f53b7a506e52ea8f581038f" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.097ex; height:2.676ex;" alt="{\displaystyle hh^{\operatorname {T} }}"></span>. </p><p>For a given relation <i>R</i>, a maximal rectangular relation contained in <i>R</i> is called a concept in <i>R</i>. Relations may be studied by decomposing into concepts, and then noting the <a href="/wiki/Heterogeneous_relation#Induced_concept_lattice" class="mw-redirect" title="Heterogeneous relation">induced concept lattice</a>. </p><p>Consider the table of group-like structures, where "unneeded" can be denoted 0, and "required" denoted by 1, forming a logical matrix <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> <mo>.</mo> </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/fdcae6b33a27f86c7961318cd7ee3d789d3bcdd2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.411ex; height:2.176ex;" alt="{\displaystyle R.}"></span> To calculate elements 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 RR^{\operatorname {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RR^{\operatorname {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729e235ef344bfc362fbf69f4424471c935d4bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.947ex; height:2.676ex;" alt="{\displaystyle RR^{\operatorname {T} }}"></span>, it is necessary to use the logical inner product of pairs of logical vectors in rows of this matrix. If this inner product is 0, then the rows are orthogonal. In fact, <a href="/wiki/Small_category" class="mw-redirect" title="Small category">small category</a> is orthogonal to <a href="/wiki/Quasigroup" title="Quasigroup">quasigroup</a>, and <a href="/wiki/Groupoid" title="Groupoid">groupoid</a> is orthogonal to <a href="/wiki/Magma_(algebra)" title="Magma (algebra)">magma</a>. Consequently there are zeros 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 RR^{\operatorname {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>R</mi> <msup> <mi>R</mi> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle RR^{\operatorname {T} }}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/729e235ef344bfc362fbf69f4424471c935d4bab" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.947ex; height:2.676ex;" alt="{\displaystyle RR^{\operatorname {T} }}"></span>, and it fails to be a <a href="/wiki/Universal_relation" class="mw-redirect" title="Universal relation">universal relation</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Row_and_column_sums">Row and column sums</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Logical_matrix&action=edit&section=7" title="Edit section: Row and column sums"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Adding up all the ones in a logical matrix may be accomplished in two ways: first summing the rows or first summing the columns. When the row sums are added, the sum is the same as when the column sums are added. In <a href="/wiki/Incidence_geometry" title="Incidence geometry">incidence geometry</a>, the matrix is interpreted as an <a href="/wiki/Incidence_matrix" title="Incidence matrix">incidence matrix</a> with the rows corresponding to "points" and the columns as "blocks" (generalizing lines made of points). A row sum is called its <i>point degree</i>, and a column sum is the <i>block degree</i>. The sum of point degrees equals the sum of block degrees.<sup id="cite_ref-BJL_5-0" class="reference"><a href="#cite_note-BJL-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>An early problem in the area was "to find necessary and sufficient conditions for the existence of an <a href="/wiki/Incidence_structure" title="Incidence structure">incidence structure</a> with given point degrees and block degrees; or in matrix language, for the existence of a (0, 1)-matrix of type <i>v</i> × <i>b</i> with given row and column sums".<sup id="cite_ref-BJL_5-1" class="reference"><a href="#cite_note-BJL-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> This problem is solved by the <a href="/wiki/Gale%E2%80%93Ryser_theorem" title="Gale–Ryser theorem">Gale–Ryser theorem</a>. </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=Logical_matrix&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/List_of_matrices" class="mw-redirect" title="List of matrices">List of matrices</a></li> <li><a href="/wiki/De_Bruijn_torus" title="De Bruijn torus">Binatorix</a> (a binary De Bruijn torus)</li> <li><a href="/wiki/Bit_array" title="Bit array">Bit array</a></li> <li><a href="/wiki/Disjunct_matrix" title="Disjunct matrix">Disjunct matrix</a></li> <li><a href="/wiki/Redheffer_matrix" title="Redheffer matrix">Redheffer matrix</a></li> <li><a href="/wiki/Truth_table" title="Truth table">Truth table</a></li> <li><a href="/wiki/Three-valued_logic" title="Three-valued logic">Three-valued logic</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=Logical_matrix&action=edit&section=9" title="Edit section: Notes"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist"> <div class="mw-references-wrap"><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"><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="CITEREFPetersen2013" class="citation web cs1">Petersen, Kjeld (February 8, 2013). <a rel="nofollow" class="external text" href="http://senseis.xmp.net/?BinMatrix">"Binmatrix"</a><span class="reference-accessdate">. Retrieved <span class="nowrap">August 11,</span> 2017</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Binmatrix&rft.date=2013-02-08&rft.aulast=Petersen&rft.aufirst=Kjeld&rft_id=http%3A%2F%2Fsenseis.xmp.net%2F%3FBinMatrix&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFO'NeilO'Neil1973" class="citation journal cs1">O'Neil, Patrick E.; <a href="/wiki/Elizabeth_O%27Neil" title="Elizabeth O'Neil">O'Neil, Elizabeth J.</a> (1973). "A Fast Expected Time Algorithm for Boolean Matrix Multiplication and Transitive Closure". <i><a href="/wiki/Information_and_Control" class="mw-redirect" title="Information and Control">Information and Control</a></i>. <b>22</b> (2): 132–8. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fs0019-9958%2873%2990228-3">10.1016/s0019-9958(73)90228-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Information+and+Control&rft.atitle=A+Fast+Expected+Time+Algorithm+for+Boolean+Matrix+Multiplication+and+Transitive+Closure&rft.volume=22&rft.issue=2&rft.pages=132-8&rft.date=1973&rft_id=info%3Adoi%2F10.1016%2Fs0019-9958%2873%2990228-3&rft.aulast=O%27Neil&rft.aufirst=Patrick+E.&rft.au=O%27Neil%2C+Elizabeth+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span> — The algorithm relies on addition being <a href="/wiki/Idempotent" class="mw-redirect" title="Idempotent">idempotent</a>, cf. p.134 (bottom).</span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCopilowish1948" class="citation journal cs1"><a href="/wiki/Irving_Copilowish" class="mw-redirect" title="Irving Copilowish">Copilowish, Irving</a> (December 1948). "Matrix development of the calculus of relations". <i><a href="/wiki/Journal_of_Symbolic_Logic" title="Journal of Symbolic Logic">Journal of Symbolic Logic</a></i>. <b>13</b> (4): 193–203. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.2307%2F2267134">10.2307/2267134</a>. <a href="/wiki/JSTOR_(identifier)" class="mw-redirect" title="JSTOR (identifier)">JSTOR</a> <a rel="nofollow" class="external text" href="https://www.jstor.org/stable/2267134">2267134</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Symbolic+Logic&rft.atitle=Matrix+development+of+the+calculus+of+relations&rft.volume=13&rft.issue=4&rft.pages=193-203&rft.date=1948-12&rft_id=info%3Adoi%2F10.2307%2F2267134&rft_id=https%3A%2F%2Fwww.jstor.org%2Fstable%2F2267134%23id-name%3DJSTOR&rft.aulast=Copilowish&rft.aufirst=Irving&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></span> </li> <li id="cite_note-GS-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-GS_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-GS_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchmidt2013" class="citation book cs1"><a href="/wiki/Gunther_Schmidt" title="Gunther Schmidt">Schmidt, Gunther</a> (2013). "6: Relations and Vectors". <i>Relational Mathematics</i>. Cambridge University Press. p. 91. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511778810">10.1017/CBO9780511778810</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-511-77881-0" title="Special:BookSources/978-0-511-77881-0"><bdi>978-0-511-77881-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=6%3A+Relations+and+Vectors&rft.btitle=Relational+Mathematics&rft.pages=91&rft.pub=Cambridge+University+Press&rft.date=2013&rft_id=info%3Adoi%2F10.1017%2FCBO9780511778810&rft.isbn=978-0-511-77881-0&rft.aulast=Schmidt&rft.aufirst=Gunther&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></span> </li> <li id="cite_note-BJL-5"><span class="mw-cite-backlink">^ <a href="#cite_ref-BJL_5-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-BJL_5-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text">E.g., see <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBethJungnickelLenz1999" class="citation encyclopaedia cs1">Beth, Thomas; <a href="/wiki/Dieter_Jungnickel" title="Dieter Jungnickel">Jungnickel, Dieter</a>; <a href="/wiki/Hanfried_Lenz" title="Hanfried Lenz">Lenz, Hanfried</a> (1999). "I. Examples and basic definitions". <i>Design Theory</i>. <i>Encyclopedia of Mathematics and its Applications</i>. Vol. 69 (2nd ed.). <a href="/wiki/Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. p. 18. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511549533.001">10.1017/CBO9780511549533.001</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-44432-3" title="Special:BookSources/978-0-521-44432-3"><bdi>978-0-521-44432-3</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Design+Theory&rft.btitle=Encyclopedia+of+Mathematics+and+its+Applications&rft.pages=18&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1999&rft_id=info%3Adoi%2F10.1017%2FCBO9780511549533.001&rft.isbn=978-0-521-44432-3&rft.aulast=Beth&rft.aufirst=Thomas&rft.au=Jungnickel%2C+Dieter&rft.au=Lenz%2C+Hanfried&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></span> </li> </ol></div></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=Logical_matrix&action=edit&section=10" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrualdi2006" class="citation encyclopaedia cs1"><a href="/wiki/Richard_A._Brualdi" title="Richard A. Brualdi">Brualdi, Richard A.</a> (2006). "Combinatorial Matrix Classes". <i>Encyclopedia of Mathematics and its Applications</i>. Vol. 108. Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511721182">10.1017/CBO9780511721182</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-86565-4" title="Special:BookSources/978-0-521-86565-4"><bdi>978-0-521-86565-4</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Combinatorial+Matrix+Classes&rft.btitle=Encyclopedia+of+Mathematics+and+its+Applications&rft.pub=Cambridge+University+Press&rft.date=2006&rft_id=info%3Adoi%2F10.1017%2FCBO9780511721182&rft.isbn=978-0-521-86565-4&rft.aulast=Brualdi&rft.aufirst=Richard+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBrualdiRyser1991" class="citation encyclopaedia cs1">Brualdi, Richard A.; Ryser, Herbert J. (1991). "Combinatorial Matrix Theory". <i>Encyclopedia of Mathematics and its Applications</i>. Vol. 39. Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9781107325708">10.1017/CBO9781107325708</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-32265-0" title="Special:BookSources/0-521-32265-0"><bdi>0-521-32265-0</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Combinatorial+Matrix+Theory&rft.btitle=Encyclopedia+of+Mathematics+and+its+Applications&rft.pub=Cambridge+University+Press&rft.date=1991&rft_id=info%3Adoi%2F10.1017%2FCBO9781107325708&rft.isbn=0-521-32265-0&rft.aulast=Brualdi&rft.aufirst=Richard+A.&rft.au=Ryser%2C+Herbert+J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBotha2013" class="citation cs2"><a href="/wiki/Leslie_Hogben" title="Leslie Hogben">Botha, J.D.</a> (2013), "31. Matrices over Finite Fields §31.3 Binary Matrices", in Hogben, Leslie (ed.), <i>Handbook of Linear Algebra (Discrete Mathematics and Its Applications)</i> (2nd ed.), Chapman & Hall/CRC, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1201%2Fb16113">10.1201/b16113</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-429-18553-3" title="Special:BookSources/978-0-429-18553-3"><bdi>978-0-429-18553-3</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=31.+Matrices+over+Finite+Fields+%C2%A731.3+Binary+Matrices&rft.btitle=Handbook+of+Linear+Algebra+%28Discrete+Mathematics+and+Its+Applications%29&rft.edition=2nd&rft.pub=Chapman+%26+Hall%2FCRC&rft.date=2013&rft_id=info%3Adoi%2F10.1201%2Fb16113&rft.isbn=978-0-429-18553-3&rft.aulast=Botha&rft.aufirst=J.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKim1982" class="citation cs2"><a href="/wiki/Ki-Hang_Kim" title="Ki-Hang Kim">Kim, Ki Hang</a> (1982), <i>Boolean Matrix Theory and Applications</i>, Dekker, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-8247-1788-9" title="Special:BookSources/978-0-8247-1788-9"><bdi>978-0-8247-1788-9</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Boolean+Matrix+Theory+and+Applications&rft.pub=Dekker&rft.date=1982&rft.isbn=978-0-8247-1788-9&rft.aulast=Kim&rft.aufirst=Ki+Hang&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyser1957" class="citation journal cs1"><a href="/wiki/H._J._Ryser" title="H. J. Ryser">Ryser, H.J.</a> (1957). "Combinatorial properties of matrices of zeroes and ones". <i><a href="/wiki/Canadian_Journal_of_Mathematics" title="Canadian Journal of Mathematics">Canadian Journal of Mathematics</a></i>. <b>9</b>: 371–7.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Mathematics&rft.atitle=Combinatorial+properties+of+matrices+of+zeroes+and+ones&rft.volume=9&rft.pages=371-7&rft.date=1957&rft.aulast=Ryser&rft.aufirst=H.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyser1960" class="citation journal cs1">Ryser, H.J. (1960). "Traces of matrices of zeroes and ones". <i>Canadian Journal of Mathematics</i>. <b>12</b>: 463–476. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-1960-040-0">10.4153/CJM-1960-040-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Mathematics&rft.atitle=Traces+of+matrices+of+zeroes+and+ones&rft.volume=12&rft.pages=463-476&rft.date=1960&rft_id=info%3Adoi%2F10.4153%2FCJM-1960-040-0&rft.aulast=Ryser&rft.aufirst=H.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRyser1960" class="citation journal cs1">Ryser, H.J. (1960). <a rel="nofollow" class="external text" href="https://www.ams.org/journals/bull/1960-66-06/S0002-9904-1960-10494-6/S0002-9904-1960-10494-6.pdf">"Matrices of Zeros and Ones"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Bulletin_of_the_American_Mathematical_Society" title="Bulletin of the American Mathematical Society">Bulletin of the American Mathematical Society</a></i>. <b>66</b>: 442–464.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Bulletin+of+the+American+Mathematical+Society&rft.atitle=Matrices+of+Zeros+and+Ones&rft.volume=66&rft.pages=442-464&rft.date=1960&rft.aulast=Ryser&rft.aufirst=H.J.&rft_id=https%3A%2F%2Fwww.ams.org%2Fjournals%2Fbull%2F1960-66-06%2FS0002-9904-1960-10494-6%2FS0002-9904-1960-10494-6.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFulkerson1960" class="citation journal cs1"><a href="/wiki/D._R._Fulkerson" title="D. R. Fulkerson">Fulkerson, D.R.</a> (1960). <a rel="nofollow" class="external text" href="https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-10/issue-3/Zero-one-matrices-with-zero-trace/pjm/1103038231.pdf">"Zero-one matrices with zero trace"</a> <span class="cs1-format">(PDF)</span>. <i><a href="/wiki/Pacific_Journal_of_Mathematics" title="Pacific Journal of Mathematics">Pacific Journal of Mathematics</a></i>. <b>10</b>: 831–6.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Pacific+Journal+of+Mathematics&rft.atitle=Zero-one+matrices+with+zero+trace&rft.volume=10&rft.pages=831-6&rft.date=1960&rft.aulast=Fulkerson&rft.aufirst=D.R.&rft_id=https%3A%2F%2Fprojecteuclid.org%2Fjournals%2Fpacific-journal-of-mathematics%2Fvolume-10%2Fissue-3%2FZero-one-matrices-with-zero-trace%2Fpjm%2F1103038231.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFulkersonRyser1961" class="citation journal cs1">Fulkerson, D.R.; Ryser, H.J. (1961). "Widths and heights of (0, 1)-matrices". <i>Canadian Journal of Mathematics</i>. <b>13</b>: 239–255. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4153%2FCJM-1961-020-3">10.4153/CJM-1961-020-3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Canadian+Journal+of+Mathematics&rft.atitle=Widths+and+heights+of+%280%2C+1%29-matrices&rft.volume=13&rft.pages=239-255&rft.date=1961&rft_id=info%3Adoi%2F10.4153%2FCJM-1961-020-3&rft.aulast=Fulkerson&rft.aufirst=D.R.&rft.au=Ryser%2C+H.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3ALogical+matrix" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFord_Jr.Fulkerson2016" class="citation book cs1"><a href="/wiki/L._R._Ford_Jr." title="L. R. Ford Jr.">Ford Jr., L.R.</a>; Fulkerson, D.R. (2016) [1962]. <a rel="nofollow" class="external text" href="https://www.degruyter.com/document/doi/10.1515/9781400875184-004/html">"II. Feasibility Theorems and Combinatorial Applications §2.12 Matrices composed of 0's and 1's"</a>. <i>Flows in Networks</i>. <a href="/wiki/Princeton_University_Press" title="Princeton University Press">Princeton University Press</a>. pp. 79–91. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1515%2F9781400875184-004">10.1515/9781400875184-004</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781400875184" title="Special:BookSources/9781400875184"><bdi>9781400875184</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0159700">0159700</a>.</cite><span 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href="/wiki/Special:EditPage/Template:Matrix_classes" title="Special:EditPage/Template:Matrix classes"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Matrix_classes" style="font-size:114%;margin:0 4em"><a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">Matrix</a> classes</div></th></tr><tr><th scope="row" class="navbox-group" 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 class="mw-selflink selflink">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 href="/wiki/Unimodular_matrix" title="Unimodular matrix">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" 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