%% This document created by Scientific Notebook (R) Version 3.0


\documentclass[12pt,thmsa]{article}
\usepackage{amssymb}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\usepackage{makeidx}
\usepackage{sw20jart}

%TCIDATA{TCIstyle=article/art4.lat,jart,sw20jart}

%TCIDATA{<META NAME="ViewSettings" CONTENT="23">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Polynomials and Vectors Associated with a Matrix">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:10:09 1997}
%TCIDATA{Language=American English}
%TCIDATA{CSTFile=MathHelp.cst}
%TCIDATA{<LINK REL="author" HREF="DM0-1.tex">}
%TCIDATA{<LINK REL="first" HREF="DM8-1.tex">}
%TCIDATA{<LINK REL="contents" HREF="DM0-0.tex">}
%TCIDATA{<LINK REL="copyright" HREF="DM0-2.tex">}
%TCIDATA{<LINK REL="end" HREF="DM8-9.tex">}
%TCIDATA{<LINK REL="index" HREF="DMIndex.tex">}
%TCIDATA{<LINK REL="next" HREF="DM8-7.tex">}
%TCIDATA{<LINK REL="parent" HREF="DM8.tex">}
%TCIDATA{<LINK REL="previous" HREF="DM8-5.tex">}
%TCIDATA{<LINK REL="top" HREF="DM0-0.tex">}

\input{tcilatex}
\begin{document}


\section{Polynomials and Vectors Associated with a Matrix}

A square matrix has characteristic and minimum polynomials, eigenvalues and
eigenvectors.

\subsection{\ Characteristic \label{Characteristic, polynomial}Polynomial
and Minimum Polynomial}

The \textsl{characteristic polynomial } 
\index{Characteristic@Characteristic!polynomial@polynomial}of a square
matrix $A$ is 
\index{Matrices@Matrices!characteristic polynomial@characteristic polynomial}%
\label{Matrices, characteristic polynomial}the determinant of the
characteristic matrix $xI-A$.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Characteristic Polynomial}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{ccc}
4 & 1 & 0 \\ 
0 & 4 & 0 \\ 
0 & 0 & 4
\end{array}
\right) $, characteristic polynomial: $\left( X-4\right) ^{3}$
\end{quotation}

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$X\left( 
\begin{array}{ccc}
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array}
\right) -\left( 
\begin{array}{ccc}
4 & 1 & 0 \\ 
0 & 4 & 0 \\ 
0 & 0 & 4
\end{array}
\right) =\,\left( 
\begin{array}{ccc}
-4+X & -1 & 0 \\ 
0 & -4+X & 0 \\ 
0 & 0 & -4+X
\end{array}
\right) $

\ 

$\det \,\left( 
\begin{array}{ccc}
-4+X & -1 & 0 \\ 
0 & -4+X & 0 \\ 
0 & 0 & -4+X
\end{array}
\right) =\,\left( X-4\right) ^{3}$
\end{quotation}

The \textsl{minimum polynomial} 
\index{Minimum@Minimum!polynomial@polynomial}\label{Minimum polynomial}of a
square matrix $A$ is the monic polynomial $p(x)$ of smallest 
\index{Matrices@Matrices!minimum polynomial@minimum polynomial}\label%
{Matrices, minimum polynomial}degree such that $p(A)=0$. By the
Cayley--Hamilton theorem 
\index{Cayley-Hamilton theorem@Cayley--Hamilton theorem}\label%
{Cayley-Hamilton Theorem}, $f(A)=0$ if $f(x)$ is the characteristic
polynomial of $A$. The minimum polynomial of $A$ is a factor of the
characteristic polynomial of $A$.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Minimum Polynomial, Factor}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{ccc}
4 & 1 & 0 \\ 
0 & 4 & 0 \\ 
0 & 0 & 4
\end{array}
\right) $, minimum polynomial: $16-8X+X^{2}=\,\left( X-4\right) ^{2}\medskip 
$
\end{quotation}

\strut

\begin{example}
This example illustrates the Cayley--Hamilton theorem.

Define $p(X)=\,X^{2}-8X+16$ and $A=\left( 
\begin{array}{ccc}
4 & 1 & 0 \\ 
0 & 4 & 0 \\ 
0 & 0 & 4
\end{array}
\right) $.

With Maple, apply \textsf{Evaluate} twice to get the following. 
\[
p(A)=\,A^{2}-8A+16=\,\left( 
\begin{array}{ccc}
0 & 0 & 0 \\ 
0 & 0 & 0 \\ 
0 & 0 & 0
\end{array}
\right) 
\]
\end{example}

The minimum and characteristic polynomial operations have to return a
variable for the polynomial. In the preceding examples, they returned $X$.
However, the variable used depends on the matrix entries and you do not need
to avoid $X$ in the matrix. This point is illustrated in the following
examples.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Minimum Polynomial}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
3X & x \\ 
5 & y
\end{array}
\right) $, minimum polynomial: $-5x+3Xy+\left( -3X-y\right) \lambda +\lambda
^{2}$

\ 

$\left( 
\begin{array}{cc}
\lambda & x \\ 
X & y
\end{array}
\right) $, characteristic polynomial: $\Theta ^{2}-\left( y+\lambda \right)
\Theta +\lambda y-xX\medskip $
\end{quotation}

\strut

\subsection{\ Eigenvalues and Eigenvectors\label{Eigenvectors}}

Given a matrix $A$, the matrix 
\index{Eigenvectors@Eigenvectors}commands \textsf{Eigenvectors } 
\index{Matrices@Matrices!eigenvectors@eigenvectors}\label%
{Matrices, eigenvectors}and \textsf{Eigenvalues} 
\index{Eigenvalues@Eigenvalues}\label{Eigenvalues} on the \textsf{Matrices}
submenu find scalars $c$ and nonzero 
\index{Matrices@Matrices!eigenvalues@eigenvalues}\label%
{Matrices, eigenvalues} vectors $v$ for which $Av=cv$. If there is a
floating-point number in the matrix, you get a numerical solution.
Otherwise, you get an exact symbolic solution.

These scalars and vectors are sometimes called \label{Characteristic value}%
\textsl{characteristic} values 
\index{Characteristic@Characteristic!vector@vector} and 
\index{Characteristic@Characteristic!value@value} \textsl{characteristic}%
\label{Characteristic vector} vectors. The eigenvalues, or characteristic
values, are roots of the characteristic polynomial.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Eigenvalues}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{rr}
\cos \alpha & -\sin \alpha \\ 
\sin \alpha & \cos \alpha
\end{array}
\right) $, eigenvalues: $\cos \alpha +i\sin \alpha ,\,\cos \alpha -i\sin
\alpha $
\end{quotation}

This matrix has characteristic polynomial $\,X^{2}-2X\cos \alpha +1$.
Replacing $X$ by the eigenvalue $\cos \alpha +i\sin \alpha $ and applying 
\textsf{Simplify} gives 
\[
\left( \cos \alpha +i\sin \alpha \right) ^{2}-2\left( \cos \alpha +i\sin
\alpha \right) \cos \alpha +1=\,\,0 
\]
demonstrating that eigenvalues are roots of the characteristic polynomial.
Note the different results obtained using integer versus floating-point
entries.\ 

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Eigenvalues}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
1 & 2 \\ 
3 & 4
\end{array}
\right) $, eigenvalues: $%
\frac{5}{2}+\frac{1}{2}\sqrt{33},\allowbreak \frac{5}{2}-\frac{1}{2}\sqrt{33}
$

\ 

$\left( 
\begin{array}{cc}
1.0 & 2 \\ 
3 & 4
\end{array}
\right) $, eigenvalues: $-.37228,5.3723$
\end{quotation}

When you choose \textsf{Eigenvectors} from the \textsf{Matrices} submenu,
with each eigenvector, you get the corresponding eigenvalue. These
eigenvectors are grouped by eigenvalues, and the multiplicity for each
eigenvalue is indicated.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Eigenvectors}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
1 & 2 \\ 
3 & 4
\end{array}
\right) $, eigenvectors: $\left\{ \left( 
\begin{array}{c}
\frac{1}{3}\rho -\frac{4}{3} \\ 
1
\end{array}
\right) \right\} \leftrightarrow \rho $ where $\rho $ is a root of $%
Z^{2}-5Z-2$
\end{quotation}

In the preceding example, $\rho $ denotes an eigenvalue, and $1$ indicates
that eigenvalue's multiplicity as a root of the characteristic polynomial.
The roots of the polynomial $Z^2-5Z-2$ are the eigenvalues computed earlier: 
$\frac 52+\frac 12\sqrt{33}$ and $\frac 52-\frac 12\sqrt{33}.$

$\ $For $\rho =\frac 52\pm \frac 12\sqrt{33}$, the corresponding eigenvector
is

\[
\left( 
\begin{array}{c}
1 \\ 
\frac 12\rho -\frac 12
\end{array}
\right) =\left( 
\begin{array}{c}
1 \\ 
\frac 12\left( \frac 52\pm \frac 12\sqrt{33}\right) -\frac 12
\end{array}
\right) =\left( 
\begin{array}{c}
1 \\ 
\frac 34\pm \frac 14\sqrt{33}
\end{array}
\right) 
\]

The products of the matrix with these eigenvectors and eigenvalues are as
follows.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
1 & 2 \\ 
3 & 4
\end{array}
\right) \left( 
\begin{array}{c}
1 \\ 
\frac{3+\sqrt{33}}{4}
\end{array}
\right) =\left( 
\begin{array}{c}
\frac{5+\sqrt{33}}{2} \\ 
6+\sqrt{33}
\end{array}
\right) ,$\quad $\frac{5+\sqrt{33}}{2}\left( 
\begin{array}{c}
1 \\ 
\frac{3+\sqrt{33}}{4}
\end{array}
\right) =\left( 
\begin{array}{c}
\frac{5}{2}+\frac{1}{2}\sqrt{33} \\ 
6+\sqrt{33}
\end{array}
\right) $

\ 

$\left( 
\begin{array}{cc}
1 & 2 \\ 
3 & 4
\end{array}
\right) \left( 
\begin{array}{c}
1 \\ 
\frac{3-\sqrt{33}}{4}
\end{array}
\right) =\left( 
\begin{array}{c}
\frac{5-\sqrt{33}}{2} \\ 
6-\sqrt{33}
\end{array}
\right) ,$\quad $\frac{5-\sqrt{33}}{2}\left( 
\begin{array}{c}
1 \\ 
\frac{3-\sqrt{33}}{4}
\end{array}
\right) =\left( 
\begin{array}{c}
\frac{5}{2}-\frac{1}{2}\sqrt{33} \\ 
6-\sqrt{33}
\end{array}
\right) $
\end{quotation}

Thus, both roots give an eigenvalue--eigenvector pair.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Eigenvectors}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{rrr}
5 & -6 & -6 \\ 
-1 & 4 & 2 \\ 
3 & -6 & -4
\end{array}
\right) $, eigenvectors: $\left\{ \left( 
\begin{array}{c}
2 \\ 
0 \\ 
1
\end{array}
\right) ,\left( 
\begin{array}{c}
2 \\ 
1 \\ 
0
\end{array}
\right) \right\} \leftrightarrow 2,\allowbreak \left\{ \left( 
\begin{array}{c}
-3 \\ 
1 \\ 
-3
\end{array}
\right) \right\} \leftrightarrow 1$
\end{quotation}

In the preceding example, $2$ is an eigenvalue occurring with multiplicity $%
2 $, and $1$ is an eigenvalue occurring with multiplicity $1$. The defining
property $Av=cv$ is illustrated by the following.

\begin{quote}
$\blacktriangleright $ \textsf{Evaluate}
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{rrr}
5 & -6 & -6 \\ 
-1 & 4 & 2 \\ 
3 & -6 & -4
\end{array}
\right) \,\left( 
\begin{array}{c}
2 \\ 
1 \\ 
0
\end{array}
\right) =\left( 
\begin{array}{c}
4 \\ 
2 \\ 
0
\end{array}
\right) $,\qquad $2\,\left( 
\begin{array}{c}
2 \\ 
1 \\ 
0
\end{array}
\right) =\left( 
\begin{array}{c}
4 \\ 
2 \\ 
0
\end{array}
\right) $

\thinspace

$\left( 
\begin{array}{rrr}
5 & -6 & -6 \\ 
-1 & 4 & 2 \\ 
3 & -6 & -4
\end{array}
\right) \left( 
\begin{array}{c}
2 \\ 
0 \\ 
1
\end{array}
\right) =\,\left( 
\begin{array}{c}
4 \\ 
0 \\ 
2
\end{array}
\right) ,\qquad 2\left( 
\begin{array}{c}
2 \\ 
0 \\ 
1
\end{array}
\right) =\left( 
\begin{array}{c}
4 \\ 
0 \\ 
2
\end{array}
\right) $
\end{quotation}

$\strut $

\subsection{\ Positive Definite Symmetric Matrices\label%
{Matrices, positive definite}\label{Matrices, definiteness tests}}

A symmetric matrix $A$ is \textsl{positive definite} if 
\index{Matrices@Matrices!positive definite@positive definite} 
\index{Matrices@Matrices!definiteness tests@definiteness tests}any of the
following equivalent conditions hold: all the eigenvalues of $A$ are
positive; the product $x^{T}Ax>0$ for all nonzero vectors $x$; or there
exists a nonsingular matrix $W$ such that $A=W^{T}W$.

A symmetric matrix $A$ is \textsl{positive semidefinite} if all the
eigenvalues of $A$ are nonnegative, is \textsl{negative definite} if all the
eigenvalues are negative, and is \textsl{negative semidefinite} if all the
eigenvalues are nonpositive.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Definiteness Tests}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{cc}
2 & -1 \\ 
-1 & 2
\end{array}
\right] $, eigenvalues: $3,\,1,\quad 
\begin{array}{l}
\limfunc{negative}\limfunc{definite},\,\text{\emph{false}}\; \\ 
\limfunc{negative}\limfunc{semidefinite},\,\text{\emph{false}} \\ 
\limfunc{positive}\limfunc{semidefinite},\,\text{\emph{true}}\; \\ 
\limfunc{positive}\limfunc{definite},\,\text{\emph{true}}
\end{array}
$

\ 

$\left[ 
\begin{array}{cc}
1 & -1 \\ 
-1 & 1
\end{array}
\right] $, eigenvalues: $0,\,2,\quad 
\begin{array}{l}
\limfunc{negative}\limfunc{definite},\emph{\,}\text{\emph{false}} \\ 
\limfunc{negative}\limfunc{semidefinite},\,\text{\emph{false}} \\ 
\limfunc{positive}\limfunc{semidefinite},\text{\thinspace \emph{true}}\; \\ 
\limfunc{positive}\limfunc{definite},\,\text{\emph{false}}
\end{array}
\medskip $
\end{quotation}

\strut

\subsection{\ Related topics}

\begin{itemize}
\item  \hyperref{Working With Matrices}{}{}{DM8-1.tex#Matrix Algebra}

\item  \hyperref{Standard Operations}{}{}{DM8-2.tex#Standard matrix
operations}

\item  \hyperref{Row Operations and Echelon Forms}{}{}{DM8-3.tex#Matrices,
row operations}

\item  \hyperref{Systems of Linear Equations}{}{}{DM8-4.tex#Systems of
equations}

\item  \hyperref{Matrix Equations}{}{}{DM8-4.tex#Matrix equations}

\item  \hyperref{Matrix Operators}{}{}{DM8-5.tex#Trace}

\item  \hyperref{Vector Spaces Associated with a Matrix}{}{}{DM8-7.tex#Row
space}

\item  \hyperref{Special Forms of Matrices}{}{}{DM8-8.tex#Smith normal form}

\item  \hyperref{Exercises and Solutions}{}{}{DM8-9.tex#Exercises}
\end{itemize}

\rule{0.67in}{0.01in}

\hyperref{\FRAME{itbpF}{0.5275in}{0.3399in}{0.0519in}{}{}{index.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid-file
"F";width 0.5275in;height 0.3399in;depth 0.0519in;original-width
36.875pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'index.wmf';file-properties "XNPEU";}}}{}{}{%
DMIndex.tex#Math Index}\quad {\small Index entries: \hyperref{Fill matrix}{}{%
}{DMIndex.tex#Fill matrix}, \hyperref{Matrices}{}{}{DMIndex.tex#Matrices}}

\end{document}
