1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
|
<html lang="en">
<head>
<title>Creating Permutation Matrices - Untitled</title>
<meta http-equiv="Content-Type" content="text/html">
<meta name="description" content="Untitled">
<meta name="generator" content="makeinfo 4.11">
<link title="Top" rel="start" href="index.html#Top">
<link rel="up" href="Basic-Usage.html#Basic-Usage" title="Basic Usage">
<link rel="prev" href="Creating-Diagonal-Matrices.html#Creating-Diagonal-Matrices" title="Creating Diagonal Matrices">
<link rel="next" href="Explicit-and-Implicit-Conversions.html#Explicit-and-Implicit-Conversions" title="Explicit and Implicit Conversions">
<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">
<p>
<a name="Creating-Permutation-Matrices"></a>
Next: <a rel="next" accesskey="n" href="Explicit-and-Implicit-Conversions.html#Explicit-and-Implicit-Conversions">Explicit and Implicit Conversions</a>,
Previous: <a rel="previous" accesskey="p" href="Creating-Diagonal-Matrices.html#Creating-Diagonal-Matrices">Creating Diagonal Matrices</a>,
Up: <a rel="up" accesskey="u" href="Basic-Usage.html#Basic-Usage">Basic Usage</a>
<hr>
</div>
<h4 class="subsection">20.1.2 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
<pre class="example"> P = eye (n) (:, q);
</pre>
<p>will create a permutation matrix - a special matrix object.
<pre class="example"> eye (n) (q, :)
</pre>
<p>will also work (and create a row permutation matrix), as well as
<pre class="example"> eye (n) (q1, q2).
</pre>
<p>For 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>
<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.
<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>
<p>Some other built-in functions can also return permutation matrices. Examples include
<dfn>inv</dfn> or <dfn>lu</dfn>.
</body></html>
|
