<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> <html> <!-- Created by GNU Texinfo 6.5, http://www.gnu.org/software/texinfo/ --> <head> <meta http-equiv="Content-Type" content="text/html; charset=utf-8"> <title>Expressions Involving Permutation Matrices (GNU Octave (version 5.1.0))</title> <meta name="description" content="Expressions Involving Permutation Matrices (GNU Octave (version 5.1.0))"> <meta name="keywords" content="Expressions Involving Permutation Matrices (GNU Octave (version 5.1.0))"> <meta name="resource-type" content="document"> <meta name="distribution" content="global"> <meta name="Generator" content="makeinfo"> <link href="index.html#Top" rel="start" title="Top"> <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index"> <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> <link href="Matrix-Algebra.html#Matrix-Algebra" rel="up" title="Matrix Algebra"> <link href="Function-Support.html#Function-Support" rel="next" title="Function Support"> <link href="Expressions-Involving-Diagonal-Matrices.html#Expressions-Involving-Diagonal-Matrices" rel="prev" title="Expressions Involving Diagonal Matrices"> <style type="text/css"> <!-- a.summary-letter {text-decoration: none} blockquote.indentedblock {margin-right: 0em} blockquote.smallindentedblock {margin-right: 0em; font-size: smaller} blockquote.smallquotation {font-size: smaller} div.display {margin-left: 3.2em} div.example {margin-left: 3.2em} div.lisp {margin-left: 3.2em} div.smalldisplay {margin-left: 3.2em} div.smallexample {margin-left: 3.2em} div.smalllisp {margin-left: 3.2em} kbd {font-style: oblique} pre.display {font-family: inherit} pre.format {font-family: inherit} pre.menu-comment {font-family: serif} pre.menu-preformatted {font-family: serif} pre.smalldisplay {font-family: inherit; font-size: smaller} pre.smallexample {font-size: smaller} pre.smallformat {font-family: inherit; font-size: smaller} pre.smalllisp {font-size: smaller} span.nolinebreak {white-space: nowrap} span.roman {font-family: initial; font-weight: normal} span.sansserif {font-family: sans-serif; font-weight: normal} ul.no-bullet {list-style: none} --> </style> <link rel="stylesheet" type="text/css" href="octave.css"> </head> <body lang="en"> <a name="Expressions-Involving-Permutation-Matrices"></a> <div class="header"> <p> Previous: <a href="Expressions-Involving-Diagonal-Matrices.html#Expressions-Involving-Diagonal-Matrices" accesskey="p" rel="prev">Expressions Involving Diagonal Matrices</a>, Up: <a href="Matrix-Algebra.html#Matrix-Algebra" accesskey="u" rel="up">Matrix Algebra</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p> </div> <hr> <a name="Expressions-Involving-Permutation-Matrices-1"></a> <h4 class="subsection">21.2.2 Expressions Involving Permutation Matrices</h4> <p>If <var>P</var> is a permutation matrix and <var>M</var> a matrix, the expression <code>P*M</code> will permute the rows of <var>M</var>. Similarly, <code>M*P</code> will yield a column permutation. Matrix division <code>P\M</code> and <code>M/P</code> can be used to do inverse permutation. </p> <p>The previously described syntax for creating permutation matrices can actually help an user to understand the connection between a permutation matrix and a permuting vector. Namely, the following holds, where <code>I = eye (n)</code> is an identity matrix: </p> <div class="example"> <pre class="example"> I(p,:) * M = (I*M) (p,:) = M(p,:) </pre></div> <p>Similarly, </p> <div class="example"> <pre class="example"> M * I(:,p) = (M*I) (:,p) = M(:,p) </pre></div> <p>The expressions <code>I(p,:)</code> and <code>I(:,p)</code> are permutation matrices. </p> <p>A permutation matrix can be transposed (or conjugate-transposed, which is the same, because a permutation matrix is never complex), inverting the permutation, or equivalently, turning a row-permutation matrix into a column-permutation one. For permutation matrices, transpose is equivalent to inversion, thus <code>P\M</code> is equivalent to <code>P'*M</code>. Transpose of a permutation matrix (or inverse) is a constant-time operation, flipping only a flag internally, and thus the choice between the two above equivalent expressions for inverse permuting is completely up to the user’s taste. </p> <p>Multiplication and division by permutation matrices works efficiently also when combined with sparse matrices, i.e., <code>P*S</code>, where <var>P</var> is a permutation matrix and <var>S</var> is a sparse matrix permutes the rows of the sparse matrix and returns a sparse matrix. The expressions <code>S*P</code>, <code>P\S</code>, <code>S/P</code> work analogically. </p> <p>Two permutation matrices can be multiplied or divided (if their sizes match), performing a composition of permutations. Also a permutation matrix can be indexed by a permutation vector (or two vectors), giving again a permutation matrix. Any other operations do not generally yield a permutation matrix and will thus trigger the implicit conversion. </p> <hr> <div class="header"> <p> Previous: <a href="Expressions-Involving-Diagonal-Matrices.html#Expressions-Involving-Diagonal-Matrices" accesskey="p" rel="prev">Expressions Involving Diagonal Matrices</a>, Up: <a href="Matrix-Algebra.html#Matrix-Algebra" accesskey="u" rel="up">Matrix Algebra</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p> </div> </body> </html>