-
Notifications
You must be signed in to change notification settings - Fork 0
/
section-13.html
192 lines (189 loc) · 13.9 KB
/
section-13.html
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
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
<!DOCTYPE html>
<!--********************************************-->
<!--* Generated from PreTeXt source *-->
<!--* on 2021-08-31T10:06:14-05:00 *-->
<!--* A recent stable commit (2020-08-09): *-->
<!--* 98f21740783f166a773df4dc83cab5293ab63a4a *-->
<!--* *-->
<!--* https://pretextbook.org *-->
<!--* *-->
<!--********************************************-->
<html lang="en-US">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Powers of a matrix (nonnegative exponents)</title>
<meta name="Keywords" content="Authored in PreTeXt">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<script src="https://sagecell.sagemath.org/embedded_sagecell.js"></script><script>window.MathJax = {
tex: {
inlineMath: [['\\(','\\)']],
tags: "none",
useLabelIds: true,
tagSide: "right",
tagIndent: ".8em",
packages: {'[+]': ['base', 'extpfeil', 'ams', 'amscd', 'newcommand', 'knowl']}
},
options: {
ignoreHtmlClass: "tex2jax_ignore",
processHtmlClass: "has_am",
renderActions: {
findScript: [10, function (doc) {
document.querySelectorAll('script[type^="math/tex"]').forEach(function(node) {
var display = !!node.type.match(/; *mode=display/);
var math = new doc.options.MathItem(node.textContent, doc.inputJax[0], display);
var text = document.createTextNode('');
node.parentNode.replaceChild(text, node);
math.start = {node: text, delim: '', n: 0};
math.end = {node: text, delim: '', n: 0};
doc.math.push(math);
});
}, '']
},
},
chtml: {
scale: 0.88,
mtextInheritFont: true
},
loader: {
load: ['input/asciimath', '[tex]/extpfeil', '[tex]/amscd', '[tex]/newcommand', '[pretext]/mathjaxknowl3.js'],
paths: {pretext: "https://pretextbook.org/js/lib"},
},
};
</script><script src="https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-chtml.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/jquery.min.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/jquery.sticky.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/jquery.espy.min.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/0.13/pretext.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/0.13/pretext_add_on.js"></script><script xmlns:svg="http://www.w3.org/2000/svg" src="https://pretextbook.org/js/lib/knowl.js"></script><!--knowl.js code controls Sage Cells within knowls--><script xmlns:svg="http://www.w3.org/2000/svg">sagecellEvalName='Evaluate (Sage)';
</script><link xmlns:svg="http://www.w3.org/2000/svg" href="https://fonts.googleapis.com/css?family=Open+Sans:400,400italic,600,600italic" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://fonts.googleapis.com/css?family=Inconsolata:400,700&subset=latin,latin-ext" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/pretext.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/pretext_add_on.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/banner_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/toc_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/knowls_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/style_default.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/colors_brown_gold.css" rel="stylesheet" type="text/css">
<link xmlns:svg="http://www.w3.org/2000/svg" href="https://pretextbook.org/css/0.31/setcolors.css" rel="stylesheet" type="text/css">
<!-- 2019-10-12: Temporary - CSS file for experiments with styling --><link xmlns:svg="http://www.w3.org/2000/svg" href="developer.css" rel="stylesheet" type="text/css">
</head>
<body class="pretext-book has-toc has-sidebar-left">
<a class="assistive" href="#content">Skip to main content</a><div xmlns:svg="http://www.w3.org/2000/svg" id="latex-macros" class="hidden-content" style="display:none">\(\def\R{{\mathbb R}}
\def\C{{\mathbb C}}
\def\Q{{\mathbb Q}}
\def\Z{{\mathbb Z}}
\def\N{{\mathbb N}}
\def\vec#1{\mathbf #1}
\newcommand{\adj}{\mathop{\mathrm{adj}}}
\newcommand{\diag}{\mathop{\mathrm{diag}}}
\newcommand{\proj}{\mathop{\mathrm{proj}}}
\newcommand{\Span}{\mathop{\mathrm{span}}}
\newcommand{\sgn}{\mathop{\mathrm{sgn}}}
\newcommand{\tr}{\mathop{\mathrm{tr}}}
\newcommand{\rowint}[2]{R_{#1} \leftrightarrow R_{#2}}
\newcommand{\rowmul}[2]{R_{#1}\gets {#2}R_{#1}}
\newcommand{\rowadd}[3]{R_{#1}\gets R_{#1}+#2R_{#3}}
\newcommand{\rowsub}[3]{R_{#1}\gets R_{#1}-#2R_{#3}}
\newcommand{\lt}{<}
\newcommand{\gt}{>}
\newcommand{\amp}{&}
\)</div>
<header id="masthead" class="smallbuttons"><div class="banner"><div class="container">
<a id="logo-link" href="http://www.umanitoba.ca" target="_blank"><img src="images/umlogo.png" alt="Logo image"></a><div class="title-container">
<h1 class="heading"><a href="mblinalg.html"><span class="title">Manitoba linear algebra</span></a></h1>
<p class="byline">Michael Doob</p>
</div>
</div></div>
<nav xmlns:svg="http://www.w3.org/2000/svg" id="primary-navbar" class="navbar"><div class="container">
<div class="navbar-top-buttons">
<button class="sidebar-left-toggle-button button active" aria-label="Show or hide table of contents sidebar">Contents</button><div class="tree-nav toolbar toolbar-divisor-3"><span class="threebuttons"><a id="previousbutton" class="previous-button toolbar-item button" href="section-12.html" title="Previous">Prev</a><a id="upbutton" class="up-button button toolbar-item" href="MatrixTheoryIntro.html" title="Up">Up</a><a id="nextbutton" class="next-button button toolbar-item" href="section-14.html" title="Next">Next</a></span></div>
</div>
<div class="navbar-bottom-buttons toolbar toolbar-divisor-4">
<button class="sidebar-left-toggle-button button toolbar-item active">Contents</button><a class="previous-button toolbar-item button" href="section-12.html" title="Previous">Prev</a><a class="up-button button toolbar-item" href="MatrixTheoryIntro.html" title="Up">Up</a><a class="next-button button toolbar-item" href="section-14.html" title="Next">Next</a>
</div>
</div></nav></header><div class="page">
<div xmlns:svg="http://www.w3.org/2000/svg" id="sidebar-left" class="sidebar" role="navigation"><div class="sidebar-content">
<nav id="toc"><ul>
<li class="link frontmatter"><a href="Frontmatter.html" data-scroll="Frontmatter"><span class="title">Title Page</span></a></li>
<li class="link"><a href="SysLinEq.html" data-scroll="SysLinEq"><span class="codenumber">1</span> <span class="title">Systems of Linear Equations</span></a></li>
<li class="link"><a href="MatrixTheoryIntro.html" data-scroll="MatrixTheoryIntro"><span class="codenumber">2</span> <span class="title">Matrix Theory</span></a></li>
<li class="link"><a href="Determinants.html" data-scroll="Determinants"><span class="codenumber">3</span> <span class="title">The Determinant</span></a></li>
<li class="link"><a href="EuclideanSpace.html" data-scroll="EuclideanSpace"><span class="codenumber">4</span> <span class="title">Vectors in Euclidean \(n\) space</span></a></li>
<li class="link"><a href="chapter-5.html" data-scroll="chapter-5"><span class="codenumber">5</span> <span class="title">Eigenvalues and eigenvectors</span></a></li>
<li class="link"><a href="LinearTransformations.html" data-scroll="LinearTransformations"><span class="codenumber">6</span> <span class="title">Linear transformations</span></a></li>
<li class="link"><a href="ExtraTopics.html" data-scroll="ExtraTopics"><span class="codenumber">7</span> <span class="title">Additional Topics</span></a></li>
</ul></nav><div class="extras"><nav><a class="pretext-link" href="https://pretextbook.org">Authored in PreTeXt</a><a href="https://www.mathjax.org"><img title="Powered by MathJax" src="https://www.mathjax.org/badge/badge.gif" alt="Powered by MathJax"></a></nav></div>
</div></div>
<main class="main"><div id="content" class="pretext-content"><section xmlns:svg="http://www.w3.org/2000/svg" class="section" id="section-13"><h2 class="heading hide-type">
<span class="type">Section</span> <span class="codenumber">2.6</span> <span class="title">Powers of a matrix (nonnegative exponents)</span>
</h2>
<section class="subsection" id="subsection-22"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">2.6.1</span> <span class="title">Computing the powers of a square matrix \(A\)</span>
</h3>
<p id="p-366">If \(A\) and \(B\) are both square matrices of order \(n\text{,}\) then \(AB\) is also a square matrix of order \(n\text{.}\) In particular, if \(A=B\text{,}\) then \(AA\) is defined. We call this matrix \(A^2\text{.}\) We also let \(A^3=AAA\text{.}\) Similarly we have higher powers of \(A\text{:}\)</p>
<div class="displaymath">
\begin{equation*}
A^n=\underbrace{A A A \cdots A A}_{n \text{ factors}}\quad
\text{ for } n=1,2,\ldots\text{.}
\end{equation*}
</div>
<p class="continuation">In addition, we define \(A^1=A\text{.}\)</p>
<p id="p-367">If \(A=\begin{bmatrix}1\amp2\\2\amp1\end{bmatrix}\) then it is straightforward to compute the powers of \(A\text{:}\)</p>
<figure class="table table-like" id="MatrixPowerExample"><figcaption><span class="type">Table</span><span class="space"> </span><span class="codenumber">2.6.1<span class="period">.</span></span><span class="space"> </span>Powers of \(A=\bigl[\begin{smallmatrix}1\amp2\\2\amp1\end{smallmatrix}\bigr]\)</figcaption><div class="tabular-box natural-width"><table class="tabular">
<tr><td class="l m b0 r0 l0 t0 lines">\(A^1=\begin{bmatrix}1\amp2\\2\amp1\end{bmatrix}\)</td></tr>
<tr><td class="l m b0 r0 l0 t0 lines">\(A^2=\begin{bmatrix}5\amp4\\4\amp5\end{bmatrix}\)</td></tr>
<tr><td class="l m b0 r0 l0 t0 lines">\(A^3=\begin{bmatrix}13\amp14\\14\amp13\end{bmatrix}\)</td></tr>
<tr><td class="l m b0 r0 l0 t0 lines">\(A^4=\begin{bmatrix}40\amp41\\41\amp40\end{bmatrix}\)</td></tr>
<tr><td class="l m b0 r0 l0 t0 lines">\(A^5=\begin{bmatrix}121\amp122\\122\amp121\end{bmatrix}\)</td></tr>
</table></div></figure><p id="p-368">It is not easy to write a general expression for \(A^n\text{.}\) When we have developed more sophisticated tools, we will be able to do so.</p></section><section class="subsection" id="LawOfExponents"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">2.6.2</span> <span class="title">The law of exponents</span>
</h3>
<p id="p-369">If \(m\) and \(n\) are positive integers, then by simply counting the factors we get the following two equations:</p>
<ul class="disc">
<li id="li-175"><p id="p-370">\(\displaystyle A^m A^n=\underbrace{A\cdots A}_{m \text{ factors}}\
\underbrace{A\cdots A}_{n \text{ factors}}=
\underbrace{A\cdots A}_{m+n\text{ factors}}=A^{m+n},\)</p></li>
<li id="li-176"><p id="p-371">\(\displaystyle (A^m)^n
=\underbrace{(\underbrace{A\cdots A}_{m \text{ factors}})
(\underbrace{A\cdots A}_{m \text{ factors}})\cdots
(\underbrace{A\cdots A}_{m \text{ factors}})}_{n \text{ times}}=A^{mn}\)</p></li>
</ul>
<p class="continuation">These two equations together are called <dfn class="terminology">the law of exponents</dfn>.</p></section><section class="subsection" id="subsection-24"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">2.6.3</span> <span class="title">What is \(A^0\text{?}\)</span>
</h3>
<p id="p-372">We want to define \(A^0\) so the the law of exponents remains valid. This says</p>
<div class="displaymath">
\begin{equation*}
A^n A^0= A^{n+0}=A^n
\end{equation*}
</div>
<p class="continuation">We observe that if \(A^0=I\text{,}\) the identity matrix, then this equation is valid. With this in mind we \(\textbf{define}\) \(A^0=I\text{.}\)</p></section><section class="subsection" id="subsection-25"><h3 class="heading hide-type">
<span class="type">Subsection</span> <span class="codenumber">2.6.4</span> <span class="title">Polynomials and powers of a matrix</span>
</h3>
<p id="p-373">Recall that polynomials are functions of the form \(p(x)=a_nx^n + a_{n-1}x^{n-1}+\cdots+a_1x+a_0\text{.}\) If we have a square matrix, we may refer to \(p(A)\text{.}\) By this we mean we substitute the matrix \(A\) for each \(x\) appearing in the polynomial. Whenever \(x^k\) appears, we substitute \(A^k\) for it and do the computations. For example, using the same \(A\) as before, if \(p(x)=x^2-2x+1\text{,}\) then</p>
<div class="displaymath">
\begin{equation*}
p(A)=A^2-2A+I=
\begin{bmatrix}5\amp4\\4\amp5\end{bmatrix}
-2\begin{bmatrix}1\amp2\\2\amp1\end{bmatrix}
+\begin{bmatrix}1\amp0\\0\amp1\end{bmatrix}
=
\begin{bmatrix}4\amp0\\0\amp4\end{bmatrix}
\end{equation*}
</div>
<p class="continuation">Now consider the matrix</p>
<div class="displaymath">
\begin{equation*}
B=\begin{bmatrix}2\amp-1\\1\amp0\end{bmatrix}
\end{equation*}
</div>
<p class="continuation">and the same polynomial \((x)=x^2-2x+1\text{.}\) In this case</p>
<div class="displaymath">
\begin{equation*}
p(B)=B^2-2B+I=
\begin{bmatrix}3\amp-2\\2\amp-1\end{bmatrix}
-2\begin{bmatrix}2\amp-1\\1\amp0\end{bmatrix}
+\begin{bmatrix}1\amp0\\0\amp1\end{bmatrix}
=
\begin{bmatrix}0\amp0\\0\amp0\end{bmatrix}
\end{equation*}
</div>
<p class="continuation">and so we get the (matrix) equation \(P(B)=\vec 0\text{.}\) When a matrix \(B\) satisfies the equation \(P(B)=0\text{,}\) we call \(B\) a <dfn class="terminology">root</dfn> of \(p(x)\text{.}\)</p></section></section></div></main>
</div>
</body>
</html>