<!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>Creating Permutation Matrices (GNU Octave (version 5.1.0))</title> <meta name="description" content="Creating Permutation Matrices (GNU Octave (version 5.1.0))"> <meta name="keywords" content="Creating 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="Basic-Usage.html#Basic-Usage" rel="up" title="Basic Usage"> <link href="Explicit-and-Implicit-Conversions.html#Explicit-and-Implicit-Conversions" rel="next" title="Explicit and Implicit Conversions"> <link href="Creating-Diagonal-Matrices.html#Creating-Diagonal-Matrices" rel="prev" title="Creating 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="Creating-Permutation-Matrices"></a> <div class="header"> <p> Next: <a href="Explicit-and-Implicit-Conversions.html#Explicit-and-Implicit-Conversions" accesskey="n" rel="next">Explicit and Implicit Conversions</a>, Previous: <a href="Creating-Diagonal-Matrices.html#Creating-Diagonal-Matrices" accesskey="p" rel="prev">Creating Diagonal Matrices</a>, Up: <a href="Basic-Usage.html#Basic-Usage" accesskey="u" rel="up">Basic Usage</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="Creating-Permutation-Matrices-1"></a> <h4 class="subsection">21.1.2 Creating Permutation Matrices</h4> <p>For creating permutation matrices, Octave does not introduce a new function, but rather overrides an existing syntax: permutation matrices can be conveniently created by indexing an identity matrix by permutation vectors. That is, if <var>q</var> is a permutation vector of length <var>n</var>, the expression </p> <div class="example"> <pre class="example"> P = eye (n) (:, q); </pre></div> <p>will create a permutation matrix - a special matrix object. </p> <div class="example"> <pre class="example">eye (n) (q, :) </pre></div> <p>will also work (and create a row permutation matrix), as well as </p> <div class="example"> <pre class="example">eye (n) (q1, q2). </pre></div> <p>For example: </p> <div class="example"> <pre class="example"> eye (4) ([1,3,2,4],:) ⇒ Permutation Matrix 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 eye (4) (:,[1,3,2,4]) ⇒ Permutation Matrix 1 0 0 0 0 0 1 0 0 1 0 0 0 0 0 1 </pre></div> <p>Mathematically, an identity matrix is both diagonal and permutation matrix. In Octave, <code>eye (n)</code> returns a diagonal matrix, because a matrix can only have one class. You can convert this diagonal matrix to a permutation matrix by indexing it by an identity permutation, as shown below. This is a special property of the identity matrix; indexing other diagonal matrices generally produces a full matrix. </p> <div class="example"> <pre class="example"> eye (3) ⇒ Diagonal Matrix 1 0 0 0 1 0 0 0 1 eye(3)(1:3,:) ⇒ Permutation Matrix 1 0 0 0 1 0 0 0 1 </pre></div> <p>Some other built-in functions can also return permutation matrices. Examples include <em>inv</em> or <em>lu</em>. </p> <hr> <div class="header"> <p> Next: <a href="Explicit-and-Implicit-Conversions.html#Explicit-and-Implicit-Conversions" accesskey="n" rel="next">Explicit and Implicit Conversions</a>, Previous: <a href="Creating-Diagonal-Matrices.html#Creating-Diagonal-Matrices" accesskey="p" rel="prev">Creating Diagonal Matrices</a>, Up: <a href="Basic-Usage.html#Basic-Usage" accesskey="u" rel="up">Basic Usage</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>