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Walsh matrix - Wikipedia
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href="/wiki/File:Walsh_16_*_Gould%27s-Morse.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Walsh_16_%2A_Gould%27s-Morse.svg/300px-Walsh_16_%2A_Gould%27s-Morse.svg.png" decoding="async" width="300" height="189" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/34/Walsh_16_%2A_Gould%27s-Morse.svg/450px-Walsh_16_%2A_Gould%27s-Morse.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/34/Walsh_16_%2A_Gould%27s-Morse.svg/600px-Walsh_16_%2A_Gould%27s-Morse.svg.png 2x" data-file-width="506" data-file-height="319" /></a><figcaption>Hadamard matrix of order 16 multiplied with a vector</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Natural_and_sequency_ordered_Walsh_16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Natural_and_sequency_ordered_Walsh_16.svg/320px-Natural_and_sequency_ordered_Walsh_16.svg.png" decoding="async" width="320" height="99" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Natural_and_sequency_ordered_Walsh_16.svg/480px-Natural_and_sequency_ordered_Walsh_16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Natural_and_sequency_ordered_Walsh_16.svg/640px-Natural_and_sequency_ordered_Walsh_16.svg.png 2x" data-file-width="1028" data-file-height="319" /></a><figcaption>Naturally ordered Hadamard matrix permuted into sequency-ordered Walsh matrix. The number of sign changes per row in the naturally ordered matrix is (0, 15, 7, 8, 3, 12, 4, 11, 1, 14, 6, 9, 2, 13, 5, 10), in the sequency-ordered matrix the number of sign changes is consecutive.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:LDU_decomposition_of_Walsh_16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/LDU_decomposition_of_Walsh_16.svg/320px-LDU_decomposition_of_Walsh_16.svg.png" decoding="async" width="320" height="75" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/84/LDU_decomposition_of_Walsh_16.svg/480px-LDU_decomposition_of_Walsh_16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/84/LDU_decomposition_of_Walsh_16.svg/640px-LDU_decomposition_of_Walsh_16.svg.png 2x" data-file-width="1364" data-file-height="319" /></a><figcaption><a href="/wiki/LU_decomposition" title="LU decomposition">LDU decomposition</a> of a Hadamard matrix. The ones in the <a href="/wiki/Triangular_matrices" class="mw-redirect" title="Triangular matrices">triangular matrices</a> form <a href="/wiki/Sierpinski_triangle" class="mw-redirect" title="Sierpinski triangle">Sierpinski triangles</a>. The entries of the <a href="/wiki/Diagonal_matrix" title="Diagonal matrix">diagonal matrix</a> are values from <a href="/wiki/Gould%27s_sequence" title="Gould's sequence">Gould's sequence</a>, with the minus signs distributed like the ones in <a href="/wiki/Thue%E2%80%93Morse_sequence" title="Thue–Morse sequence">Thue–Morse sequence</a>.</figcaption></figure> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Binary_Walsh_matrix_16.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Binary_Walsh_matrix_16.svg/320px-Binary_Walsh_matrix_16.svg.png" decoding="async" width="320" height="320" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Binary_Walsh_matrix_16.svg/480px-Binary_Walsh_matrix_16.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c0/Binary_Walsh_matrix_16.svg/640px-Binary_Walsh_matrix_16.svg.png 2x" data-file-width="425" data-file-height="425" /></a><figcaption>Binary Hadamard matrix as a <a href="/wiki/Matrix_multiplication" title="Matrix multiplication">matrix product</a>. The binary matrix (white 0, red 1) is the result with operations in <a href="/wiki/Finite_field" title="Finite field"><b>F</b><sub>2</sub></a>. The gray numbers show the result with operations 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 \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf9a96b565ea202d0f4322e9195613fb26a9bed" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \mathbb {N} }"></span>.</figcaption></figure> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="/wiki/Hadamard_matrix" title="Hadamard matrix">Hadamard matrix</a></div> <p>In <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, a <b>Walsh matrix</b> is a specific <a href="/wiki/Square_matrix" title="Square matrix">square matrix</a> of dimensions 2<sup><i>n</i></sup>, where <i>n</i> is some particular <a href="/wiki/Natural_number" title="Natural number">natural number</a>. The entries of the <a href="/wiki/Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> are either +1 or −1 and its rows as well as columns are <a href="/wiki/Orthogonal" class="mw-redirect" title="Orthogonal">orthogonal</a>. The Walsh matrix was proposed by <a href="/wiki/Joseph_L._Walsh" title="Joseph L. Walsh">Joseph L. Walsh</a> in 1923.<sup id="cite_ref-kanjilal_1-0" class="reference"><a href="#cite_note-kanjilal-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Each row of a Walsh matrix corresponds to a <a href="/wiki/Walsh_function" title="Walsh function">Walsh function</a>. </p><p>The Walsh matrices are a special case of <a href="/wiki/Hadamard_matrices" class="mw-redirect" title="Hadamard matrices">Hadamard matrices</a> where the rows are rearranged so that the number of sign changes in a row is in increasing order. In short, a Hadamard matrix is defined by the <a href="/wiki/Recursion" title="Recursion">recursive</a> formula below and is <i>naturally ordered</i>, whereas a Walsh matrix is <i>sequency-ordered</i>.<sup id="cite_ref-kanjilal_1-1" class="reference"><a href="#cite_note-kanjilal-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> Confusingly, different sources refer to either matrix as the Walsh matrix. </p><p>The Walsh matrix (and Walsh functions) are used in computing the <a href="/wiki/Walsh_transform" class="mw-redirect" title="Walsh transform">Walsh transform</a> and have applications in the efficient implementation of certain <a href="/wiki/Signal_processing" title="Signal processing">signal processing</a> operations. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Formula">Formula</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Walsh_matrix&action=edit&section=1" title="Edit section: Formula"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The Hadamard matrices of dimension <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 2^{k}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{k}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d82641ae2702b0db07dd11830af27b9ee0cd196" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.251ex; height:2.676ex;" alt="{\displaystyle 2^{k}}"></span> for <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k\in \mathbb {N} }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>k</mi> <mo>∈<!-- ∈ --></mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle k\in \mathbb {N} }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a5bc4b7383031ba693b7433198ead7170954c1d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.73ex; height:2.176ex;" alt="{\displaystyle k\in \mathbb {N} }"></span> are given by the recursive formula (the lowest order of Hadamard matrix is 2): </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{aligned}H\left(2^{1}\right)&={\begin{bmatrix}1&1\\1&-1\end{bmatrix}},\\H\left(2^{2}\right)&={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\\\end{bmatrix}},\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>)</mo> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}H\left(2^{1}\right)&={\begin{bmatrix}1&1\\1&-1\end{bmatrix}},\\H\left(2^{2}\right)&={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\\\end{bmatrix}},\end{aligned}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9da599cb5c04965aac8c81489509bab866d29cd6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.838ex; width:32.188ex; height:18.843ex;" alt="{\displaystyle {\begin{aligned}H\left(2^{1}\right)&={\begin{bmatrix}1&1\\1&-1\end{bmatrix}},\\H\left(2^{2}\right)&={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\\\end{bmatrix}},\end{aligned}}}"></span></dd></dl> <p>and in general </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 H\left(2^{k}\right)={\begin{bmatrix}H\left(2^{k-1}\right)&H\left(2^{k-1}\right)\\H\left(2^{k-1}\right)&-H\left(2^{k-1}\right)\end{bmatrix}}=H(2)\otimes H\left(2^{k-1}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mtd> <mtd> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mtd> <mtd> <mo>−<!-- − --></mo> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>=</mo> <mi>H</mi> <mo stretchy="false">(</mo> <mn>2</mn> <mo stretchy="false">)</mo> <mo>⊗<!-- ⊗ --></mo> <mi>H</mi> <mrow> <mo>(</mo> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mo>−<!-- − --></mo> <mn>1</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H\left(2^{k}\right)={\begin{bmatrix}H\left(2^{k-1}\right)&H\left(2^{k-1}\right)\\H\left(2^{k-1}\right)&-H\left(2^{k-1}\right)\end{bmatrix}}=H(2)\otimes H\left(2^{k-1}\right),}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/916e5102b880b693b01fcb6929985006ad7e2244" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:56.328ex; height:7.509ex;" alt="{\displaystyle H\left(2^{k}\right)={\begin{bmatrix}H\left(2^{k-1}\right)&H\left(2^{k-1}\right)\\H\left(2^{k-1}\right)&-H\left(2^{k-1}\right)\end{bmatrix}}=H(2)\otimes H\left(2^{k-1}\right),}"></span></dd></dl> <p>for 2 ≤ <i>k</i> ∈ <b>N</b>, where ⊗ denotes the <a href="/wiki/Kronecker_product" title="Kronecker product">Kronecker product</a>. </p> <div class="mw-heading mw-heading3"><h3 id="Permutation">Permutation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Walsh_matrix&action=edit&section=2" title="Edit section: Permutation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>We can obtain a Walsh matrix from a Hadamard matrix. For that, we first generate the Hadamard matrix for a given dimension. Then, we count the number of sign changes of each row. Finally, we re-order the rows of the matrix according to the number of sign changes in ascending order. </p><p>For example, let us assume that we have a Hadamard matrix of dimension <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 2^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mn>2</mn> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle 2^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/efd7711cd907a2d46557a410fb67fc0d84c52ba3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.217ex; height:2.676ex;" alt="{\displaystyle 2^{2}}"></span> </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 H(4)={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\\\end{bmatrix}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(4)={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\\\end{bmatrix}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bb67a4d557226b961e7bcf35b82c17b7bad236d8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:29.028ex; height:12.509ex;" alt="{\displaystyle H(4)={\begin{bmatrix}1&1&1&1\\1&-1&1&-1\\1&1&-1&-1\\1&-1&-1&1\\\end{bmatrix}}}"></span>,</dd></dl> <p>where the successive rows have 0, 3, 1, and 2 sign changes (we count the number of times we switch from a positive 1 to a negative 1, and vice versa). If we rearrange the rows in sequency ordering, we obtain: </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 W(4)={\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&-1&1\\1&-1&1&-1\\\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mn>4</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(4)={\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&-1&1\\1&-1&1&-1\\\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11d1b3dc4d3b23dfa73150f9a6f7adb2c2f211fc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.671ex; width:30.046ex; height:12.509ex;" alt="{\displaystyle W(4)={\begin{bmatrix}1&1&1&1\\1&1&-1&-1\\1&-1&-1&1\\1&-1&1&-1\\\end{bmatrix}},}"></span></dd></dl> <p>where the successive rows have 0, 1, 2, and 3 sign changes. </p> <div class="mw-heading mw-heading3"><h3 id="Alternative_forms_of_the_Walsh_matrix">Alternative forms of the Walsh matrix</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Walsh_matrix&action=edit&section=3" title="Edit section: Alternative forms of the Walsh matrix"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <div class="mw-heading mw-heading4"><h4 id="Sequency_ordering">Sequency ordering</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Walsh_matrix&action=edit&section=4" title="Edit section: Sequency ordering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The sequency ordering of the rows of the Walsh matrix can be derived from the ordering of the Hadamard matrix by first applying the <a href="/wiki/Bit-reversal_permutation" title="Bit-reversal permutation">bit-reversal permutation</a> and then the <a href="/wiki/Gray_code" title="Gray code">Gray-code</a> <a href="/wiki/Permutation" title="Permutation">permutation</a>:<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> <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 W(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&-1&-1&1&1\\1&1&-1&-1&1&1&-1&-1\\1&-1&-1&1&1&-1&-1&1\\1&-1&-1&1&-1&1&1&-1\\1&-1&1&-1&-1&1&-1&1\\1&-1&1&-1&1&-1&1&-1\\\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&-1&-1&1&1\\1&1&-1&-1&1&1&-1&-1\\1&-1&-1&1&1&-1&-1&1\\1&-1&-1&1&-1&1&1&-1\\1&-1&1&-1&-1&1&-1&1\\1&-1&1&-1&1&-1&1&-1\\\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2d8182bd22e3417054a4eb3c210af7f80edc5099" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:51.219ex; height:25.509ex;" alt="{\displaystyle W(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&-1&-1&1&1\\1&1&-1&-1&1&1&-1&-1\\1&-1&-1&1&1&-1&-1&1\\1&-1&-1&1&-1&1&1&-1\\1&-1&1&-1&-1&1&-1&1\\1&-1&1&-1&1&-1&1&-1\\\end{bmatrix}},}"></span></dd></dl> <p>where the successive rows have 0, 1, 2, 3, 4, 5, 6, and 7 sign changes. </p> <div class="mw-heading mw-heading4"><h4 id="Dyadic_ordering">Dyadic ordering</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Walsh_matrix&action=edit&section=5" title="Edit section: Dyadic ordering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 W(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&1&1&-1&-1\\1&1&-1&-1&-1&-1&1&1\\1&-1&1&-1&1&-1&1&-1\\1&-1&1&-1&-1&1&-1&1\\1&-1&-1&1&1&-1&-1&1\\1&-1&-1&1&-1&1&1&-1\\\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>W</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle W(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&1&1&-1&-1\\1&1&-1&-1&-1&-1&1&1\\1&-1&1&-1&1&-1&1&-1\\1&-1&1&-1&-1&1&-1&1\\1&-1&-1&1&1&-1&-1&1\\1&-1&-1&1&-1&1&1&-1\\\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4975870984d69e0d92a3a155b093e02164b0c812" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:51.219ex; height:25.509ex;" alt="{\displaystyle W(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&1&1&1&-1&-1&-1&-1\\1&1&-1&-1&1&1&-1&-1\\1&1&-1&-1&-1&-1&1&1\\1&-1&1&-1&1&-1&1&-1\\1&-1&1&-1&-1&1&-1&1\\1&-1&-1&1&1&-1&-1&1\\1&-1&-1&1&-1&1&1&-1\\\end{bmatrix}},}"></span></dd></dl> <p>where the successive rows have 0, 1, 3, 2, 7, 6, 4, and 5 sign changes. </p> <div class="mw-heading mw-heading4"><h4 id="Natural_ordering">Natural ordering</h4><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Walsh_matrix&action=edit&section=6" title="Edit section: Natural ordering"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <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 H(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&-1&1&-1&1&-1&1&-1\\1&1&-1&-1&1&1&-1&-1\\1&-1&-1&1&1&-1&-1&1\\1&1&1&1&-1&-1&-1&-1\\1&-1&1&-1&-1&1&-1&1\\1&1&-1&-1&-1&-1&1&1\\1&-1&-1&1&-1&1&1&-1\\\end{bmatrix}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo stretchy="false">(</mo> <mn>8</mn> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow> <mo>[</mo> <mtable rowspacing="4pt" columnspacing="1em"> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> </mtr> <mtr> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mn>1</mn> </mtd> <mtd> <mo>−<!-- − --></mo> <mn>1</mn> </mtd> </mtr> </mtable> <mo>]</mo> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&-1&1&-1&1&-1&1&-1\\1&1&-1&-1&1&1&-1&-1\\1&-1&-1&1&1&-1&-1&1\\1&1&1&1&-1&-1&-1&-1\\1&-1&1&-1&-1&1&-1&1\\1&1&-1&-1&-1&-1&1&1\\1&-1&-1&1&-1&1&1&-1\\\end{bmatrix}},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/23fc5b564d3afd994bffe76dbbcb29e99c315037" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -12.171ex; width:50.848ex; height:25.509ex;" alt="{\displaystyle H(8)={\begin{bmatrix}1&1&1&1&1&1&1&1\\1&-1&1&-1&1&-1&1&-1\\1&1&-1&-1&1&1&-1&-1\\1&-1&-1&1&1&-1&-1&1\\1&1&1&1&-1&-1&-1&-1\\1&-1&1&-1&-1&1&-1&1\\1&1&-1&-1&-1&-1&1&1\\1&-1&-1&1&-1&1&1&-1\\\end{bmatrix}},}"></span></dd></dl> <p>where the successive rows have 0, 7, 3, 4, 1, 6, 2, and 5 sign changes (Hadamard matrix). </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=Walsh_matrix&action=edit&section=7" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1235681985">.mw-parser-output .side-box{margin:4px 0;box-sizing:border-box;border:1px solid #aaa;font-size:88%;line-height:1.25em;background-color:var(--background-color-interactive-subtle,#f8f9fa);display:flow-root}.mw-parser-output .side-box-abovebelow,.mw-parser-output .side-box-text{padding:0.25em 0.9em}.mw-parser-output .side-box-image{padding:2px 0 2px 0.9em;text-align:center}.mw-parser-output .side-box-imageright{padding:2px 0.9em 2px 0;text-align:center}@media(min-width:500px){.mw-parser-output .side-box-flex{display:flex;align-items:center}.mw-parser-output .side-box-text{flex:1;min-width:0}}@media(min-width:720px){.mw-parser-output .side-box{width:238px}.mw-parser-output .side-box-right{clear:right;float:right;margin-left:1em}.mw-parser-output .side-box-left{margin-right:1em}}</style><style data-mw-deduplicate="TemplateStyles:r1237033735">@media print{body.ns-0 .mw-parser-output .sistersitebox{display:none!important}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sistersitebox img[src*="Wiktionary-logo-en-v2.svg"]{background-color:white}}</style><div class="side-box side-box-right plainlinks sistersitebox"><style data-mw-deduplicate="TemplateStyles:r1126788409">.mw-parser-output .plainlist ol,.mw-parser-output .plainlist ul{line-height:inherit;list-style:none;margin:0;padding:0}.mw-parser-output .plainlist ol li,.mw-parser-output .plainlist ul li{margin-bottom:0}</style> <div class="side-box-flex"> <div class="side-box-image"><span class="noviewer" typeof="mw:File"><span><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/30px-Commons-logo.svg.png" decoding="async" width="30" height="40" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/45px-Commons-logo.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/4/4a/Commons-logo.svg/59px-Commons-logo.svg.png 2x" data-file-width="1024" data-file-height="1376" /></span></span></div> <div class="side-box-text plainlist">Wikimedia Commons has media related to <span style="font-weight: bold; font-style: italic;"><a href="https://commons.wikimedia.org/wiki/Category:Walsh_matrix" class="extiw" title="commons:Category:Walsh matrix">Walsh matrix</a></span>.</div></div> </div> <ul><li><a href="/wiki/Haar_wavelet" title="Haar wavelet">Haar wavelet</a></li> <li><a href="/wiki/Quincunx_matrix" title="Quincunx matrix">Quincunx matrix</a></li> <li><a href="/wiki/Hadamard_transform" title="Hadamard transform">Hadamard transform</a></li> <li><a href="/wiki/Code-division_multiple_access" title="Code-division multiple access">Code-division multiple access</a></li> <li><span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A228539" class="extiw" title="oeis:A228539">A228539</a></span> (<span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A228540" class="extiw" title="oeis:A228540">A228540</a></span>) – rows of the (negated) binary Walsh matrices read as reverse binary numbers</li> <li><span class="nowrap external"><a href="/wiki/On-Line_Encyclopedia_of_Integer_Sequences" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>: <a href="//oeis.org/A197818" class="extiw" title="oeis:A197818">A197818</a></span> – antidiagonals of the negated binary Walsh matrix read as binary numbers</li></ul> <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=Walsh_matrix&action=edit&section=8" title="Edit section: References"><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-kanjilal-1"><span class="mw-cite-backlink">^ <a href="#cite_ref-kanjilal_1-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-kanjilal_1-1"><sup><i><b>b</b></i></sup></a></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="CITEREFKanjilal1995" class="citation book cs1">Kanjilal, P. P. (1995). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=h5H5voqjNIQC&pg=PA210"><i>Adaptive Prediction and Predictive Control</i></a>. Stevenage: IET. p. 210. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-86341-193-2" title="Special:BookSources/0-86341-193-2"><bdi>0-86341-193-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Adaptive+Prediction+and+Predictive+Control&rft.place=Stevenage&rft.pages=210&rft.pub=IET&rft.date=1995&rft.isbn=0-86341-193-2&rft.aulast=Kanjilal&rft.aufirst=P.+P.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dh5H5voqjNIQC%26pg%3DPA210&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWalsh+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="CITEREFYuen1972" class="citation journal cs1">Yuen, C.-K. (1972). "Remarks on the Ordering of Walsh Functions". <i>IEEE Transactions on Computers</i>. <b>21</b> (12): 1452. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FT-C.1972.223524">10.1109/T-C.1972.223524</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=IEEE+Transactions+on+Computers&rft.atitle=Remarks+on+the+Ordering+of+Walsh+Functions&rft.volume=21&rft.issue=12&rft.pages=1452&rft.date=1972&rft_id=info%3Adoi%2F10.1109%2FT-C.1972.223524&rft.aulast=Yuen&rft.aufirst=C.-K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AWalsh+matrix" class="Z3988"></span></span> </li> </ol></div></div> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist 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matrix">Anti-diagonal</a></li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Anti-Hermitian</a></li> <li><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Anti-symmetric</a></li> <li><a href="/wiki/Arrowhead_matrix" title="Arrowhead matrix">Arrowhead</a></li> <li><a href="/wiki/Band_matrix" title="Band matrix">Band</a></li> <li><a href="/wiki/Bidiagonal_matrix" title="Bidiagonal matrix">Bidiagonal</a></li> <li><a href="/wiki/Bisymmetric_matrix" title="Bisymmetric matrix">Bisymmetric</a></li> <li><a href="/wiki/Block-diagonal_matrix" class="mw-redirect" title="Block-diagonal matrix">Block-diagonal</a></li> <li><a href="/wiki/Block_matrix" title="Block matrix">Block</a></li> <li><a href="/wiki/Block_tridiagonal_matrix" class="mw-redirect" title="Block tridiagonal matrix">Block tridiagonal</a></li> <li><a href="/wiki/Boolean_matrix" title="Boolean matrix">Boolean</a></li> <li><a href="/wiki/Cauchy_matrix" title="Cauchy matrix">Cauchy</a></li> <li><a href="/wiki/Centrosymmetric_matrix" title="Centrosymmetric matrix">Centrosymmetric</a></li> <li><a href="/wiki/Conference_matrix" title="Conference matrix">Conference</a></li> <li><a href="/wiki/Complex_Hadamard_matrix" title="Complex Hadamard matrix">Complex Hadamard</a></li> <li><a href="/wiki/Copositive_matrix" title="Copositive matrix">Copositive</a></li> <li><a href="/wiki/Diagonally_dominant_matrix" title="Diagonally dominant matrix">Diagonally dominant</a></li> <li><a href="/wiki/Diagonal_matrix" title="Diagonal matrix">Diagonal</a></li> <li><a href="/wiki/DFT_matrix" title="DFT matrix">Discrete Fourier Transform</a></li> <li><a href="/wiki/Elementary_matrix" title="Elementary matrix">Elementary</a></li> <li><a href="/wiki/Equivalent_matrix" class="mw-redirect" title="Equivalent matrix">Equivalent</a></li> <li><a href="/wiki/Frobenius_matrix" title="Frobenius matrix">Frobenius</a></li> <li><a href="/wiki/Generalized_permutation_matrix" title="Generalized permutation matrix">Generalized permutation</a></li> <li><a href="/wiki/Hadamard_matrix" title="Hadamard matrix">Hadamard</a></li> <li><a href="/wiki/Hankel_matrix" title="Hankel matrix">Hankel</a></li> <li><a href="/wiki/Hermitian_matrix" title="Hermitian matrix">Hermitian</a></li> <li><a href="/wiki/Hessenberg_matrix" title="Hessenberg matrix">Hessenberg</a></li> <li><a href="/wiki/Hollow_matrix" title="Hollow matrix">Hollow</a></li> <li><a href="/wiki/Integer_matrix" title="Integer matrix">Integer</a></li> <li><a href="/wiki/Logical_matrix" title="Logical matrix">Logical</a></li> <li><a href="/wiki/Matrix_unit" title="Matrix unit">Matrix unit</a></li> <li><a href="/wiki/Metzler_matrix" title="Metzler matrix">Metzler</a></li> <li><a href="/wiki/Moore_matrix" title="Moore matrix">Moore</a></li> <li><a href="/wiki/Nonnegative_matrix" title="Nonnegative matrix">Nonnegative</a></li> <li><a href="/wiki/Pentadiagonal_matrix" class="mw-redirect" title="Pentadiagonal matrix">Pentadiagonal</a></li> <li><a href="/wiki/Permutation_matrix" title="Permutation matrix">Permutation</a></li> <li><a href="/wiki/Persymmetric_matrix" title="Persymmetric matrix">Persymmetric</a></li> <li><a href="/wiki/Polynomial_matrix" title="Polynomial matrix">Polynomial</a></li> <li><a href="/wiki/Quaternionic_matrix" title="Quaternionic matrix">Quaternionic</a></li> <li><a href="/wiki/Signature_matrix" title="Signature matrix">Signature</a></li> <li><a href="/wiki/Skew-Hermitian_matrix" title="Skew-Hermitian matrix">Skew-Hermitian</a></li> <li><a href="/wiki/Skew-symmetric_matrix" title="Skew-symmetric matrix">Skew-symmetric</a></li> <li><a href="/wiki/Skyline_matrix" title="Skyline matrix">Skyline</a></li> <li><a href="/wiki/Sparse_matrix" title="Sparse matrix">Sparse</a></li> <li><a href="/wiki/Sylvester_matrix" title="Sylvester matrix">Sylvester</a></li> <li><a href="/wiki/Symmetric_matrix" title="Symmetric matrix">Symmetric</a></li> <li><a href="/wiki/Toeplitz_matrix" title="Toeplitz matrix">Toeplitz</a></li> <li><a href="/wiki/Triangular_matrix" title="Triangular matrix">Triangular</a></li> <li><a href="/wiki/Tridiagonal_matrix" title="Tridiagonal matrix">Tridiagonal</a></li> <li><a href="/wiki/Vandermonde_matrix" title="Vandermonde matrix">Vandermonde</a></li> <li><a class="mw-selflink selflink">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" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/24px-Nuvola_apps_edu_mathematics_blue-p.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/3e/Nuvola_apps_edu_mathematics_blue-p.svg/32px-Nuvola_apps_edu_mathematics_blue-p.svg.png 2x" data-file-width="128" data-file-height="128" /></a></span> </span><a href="/wiki/Portal:Mathematics" title="Portal:Mathematics">Mathematics portal</a></b></li> <li><a href="/wiki/List_of_matrices" class="mw-redirect" title="List of matrices">List of matrices</a></li> <li><a href="/wiki/Category:Matrices" title="Category:Matrices">Category:Matrices</a></li></ul> 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