<html lang="en"> <head> <title>Expressions Involving Permutation Matrices - GNU Octave</title> <meta http-equiv="Content-Type" content="text/html"> <meta name="description" content="GNU Octave"> <meta name="generator" content="makeinfo 4.13"> <link title="Top" rel="start" href="index.html#Top"> <link rel="up" href="Matrix-Algebra.html#Matrix-Algebra" title="Matrix Algebra"> <link rel="prev" href="Expressions-Involving-Diagonal-Matrices.html#Expressions-Involving-Diagonal-Matrices" title="Expressions Involving Diagonal Matrices"> <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> <meta http-equiv="Content-Style-Type" content="text/css"> <style type="text/css"><!-- pre.display { font-family:inherit } pre.format { font-family:inherit } pre.smalldisplay { font-family:inherit; font-size:smaller } pre.smallformat { font-family:inherit; font-size:smaller } pre.smallexample { font-size:smaller } pre.smalllisp { font-size:smaller } span.sc { font-variant:small-caps } span.roman { font-family:serif; font-weight:normal; } span.sansserif { font-family:sans-serif; font-weight:normal; } --></style> </head> <body> <div class="node"> <a name="Expressions-Involving-Permutation-Matrices"></a> <p> Previous: <a rel="previous" accesskey="p" href="Expressions-Involving-Diagonal-Matrices.html#Expressions-Involving-Diagonal-Matrices">Expressions Involving Diagonal Matrices</a>, Up: <a rel="up" accesskey="u" href="Matrix-Algebra.html#Matrix-Algebra">Matrix Algebra</a> <hr> </div> <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>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: <pre class="example"> I(p,:) * M = (I*M) (p,:) = M(p,:) </pre> <p>Similarly, <pre class="example"> M * I(:,p) = (M*I) (:,p) = M(:,p) </pre> <p>The expressions <code>I(p,:)</code> and <code>I(:,p)</code> are permutation matrices. <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>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>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. </body></html>