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%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Thu Feb 13 17:09:36 1997}
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\begin{document}


\section{Matrix Operators\label{Trace}}

A matrix operator is a function that operates on matrices.

\subsection{\ Trace}

The \textsl{trace} of an $n\times n$ matrix is \label{Matrices, trace}the
sum of the diagonal elements. This operation applies to square matrices only.

\begin{quote}
$\blacktriangleright $ To compute the trace of a square matrix
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in the matrix.

\item[2.]  From the \textsf{Matrices} submenu, choose \textsf{Trace}.
\end{enumerate}

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

\begin{quotation}
$\left( 
\begin{array}{cc}
a & b \\ 
c & d
\end{array}
\right) $, trace: $a+d\smallskip $

$\left( 
\begin{array}{ccc}
-85 & -55 & -37 \\ 
-35 & 97 & 50 \\ 
79 & 56 & 49
\end{array}
\right) $, trace: $61\bigskip $
\end{quotation}

\subsection{\ Transpose\label{Transpose}}

The \textsl{transpose} of \label{Matrices, transpose}an $m\times n$ matrix
is the $n\times m$ matrix that you obtain from the first matrix by
interchanging the rows and columns.

\begin{quote}
$\blacktriangleright $ To compute the transpose of a matrix
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in the matrix.

\item[2.]  From the \textsf{Matrices} submenu, choose \textsf{Transpose}.
\end{enumerate}

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

\begin{quotation}
$\left( 
\begin{array}{cc}
a & b \\ 
c & d
\end{array}
\right) $, transpose: $\left( 
\begin{array}{cc}
a & c \\ 
b & d
\end{array}
\right) $
\end{quotation}

\ 

You can also get the transpose of a matrix by using the superscript $T$.

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

\begin{quotation}
$\left( 
\begin{array}{ccc}
a & b & c \\ 
d & e & f
\end{array}
\right) ^{T}=\,\left( 
\begin{array}{cc}
a & d \\ 
b & e \\ 
c & f
\end{array}
\right) $\bigskip
\end{quotation}

\subsection{\ Hermitian Transpose\label{Transpose, Hermitian}}

The \textsl{Hermitian transpose} of a matrix is the transpose together with
the replacement of each entry by its complex conjugate. It is also referred
to as \label{Adjoint, Hermitian}the \textsl{adjoint} or \textsl{Hermitian
adjoint} of a matrix (not to be confused \label{Hermitian transpose}with 
\label{Matrices, Hermitian transpose}the classical adjoint or adjugate,
discussed elsewhere in this chapter.)

\begin{quote}
$\blacktriangleright $ To compute the Hermitian transpose of a matrix
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in the matrix.

\item[2.]  From the \textsf{Matrices} submenu, choose \textsf{Hermitian
Transpose}.
\end{enumerate}

\begin{quotation}
$\left( 
\begin{array}{cc}
a+ib & c+id \\ 
e+if & g+ih
\end{array}
\right) $, Hermitian transpose: $\left( 
\begin{array}{cc}
a-ib & e-if \\ 
c-id & g-ih
\end{array}
\right) $

\ 

$\left( 
\begin{array}{cc}
2+i & -i \\ 
4-i & 2+i
\end{array}
\right) $, Hermitian transpose: $\left( 
\begin{array}{cc}
2-i & 4+i \\ 
i & 2-i
\end{array}
\right) $
\end{quotation}

You can also get the Hermitian transpose of a matrix by using the
superscript $H$.

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

\begin{quotation}
$\left( 
\begin{array}{rr}
i & 2+i \\ 
4i & 3-2i
\end{array}
\right) ^{H}=\,\left( 
\begin{array}{cc}
-i & -4i \\ 
2-i & 3+2i
\end{array}
\right) \bigskip $
\end{quotation}

\subsection{\ Determinant\label{Determinant}}

\begin{quotation}
The \textsl{determinant} of \label{Matrices, determinant}an $n\times n$
matrix $\left( a_{ij}\right) $ is the sum and difference of certain products
of the entries. Specifically, 
\[
\det (a_{ij})=\sum_{\sigma }\left( -1\right) ^{sgn\left( \sigma \right)
}a_{1\sigma \left( 1\right) }a_{2\sigma \left( 2\right) }\cdots a_{n\sigma
\left( n\right) } 
\]
where $\sigma $ ranges over all the permutations of $\left\{ 1,2,\ldots
,n\right\} $ and $\left( -1\right) ^{sgn\left( \sigma \right) }=\pm 1$,
depending on whether $\sigma $ is an even or odd permutation.
\end{quotation}

Note that this operation applies to square matrices only.

\begin{quote}
$\blacktriangleright $ To compute the determinant of a square matrix
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in the matrix.

\item[2.]  From the \textsf{Matrices} submenu, choose \textsf{Determinant}.
\end{enumerate}

\begin{quotation}
$\left[ 
\begin{array}{cc}
a & b \\ 
c & d
\end{array}
\right] $, determinant: $ad-bc$

\vspace{1pt}

$\left[ 
\begin{array}{ccc}
a_{1,1} & a_{1,2} & a_{1,3} \\ 
a_{2,1} & a_{2,2} & a_{2,3} \\ 
a_{3,1} & a_{3,2} & a_{3,3}
\end{array}
\right] $, determinant: $
\begin{array}{c}
a_{1,1}a_{2,2}a_{3,3}-a_{1,1}a_{2,3}a_{3,2}-a_{2,1}a_{1,2}a_{3,3} \\ 
\quad +a_{2,1}a_{1,3}a_{3,2}+a_{3,1}a_{1,2}a_{2,3}-a_{3,1}a_{1,3}a_{2,2}
\end{array}
$

\vspace{1pt}

$\left[ 
\begin{array}{rrr}
-85 & -55 & -37 \\ 
-35 & 97 & 50 \\ 
79 & 56 & 49
\end{array}
\right] $, determinant: $-121529$
\end{quotation}

You can also denote the determinant by 
\[
\det \left[ 
\begin{array}{rrr}
-85 & -55 & -37 \\ 
-35 & 97 & 50 \\ 
79 & 56 & 49
\end{array}
\right] 
\]
where \QTR{UserInput}{det} is typed while in mathematics mode. This string
is a multicharacter function name. When typed in mathematics, it turns gray
when the $t$ is typed. This function can also be chosen from the list under 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{function.wmf}{\special%
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"1";cropbottom "0";filename 'FUNCTION.WMF';file-properties "XNPEU";}}.

\subsection{\ Adjugate\label{Adjugate}}

The \emph{adjugate} or \emph{classical adjoint}\textsl{\ } \label%
{Adjoint, classical}of a matrix $A$ is the transpose of the matrix \label%
{Matrices, adjugate}of cofactors of $A$. The $i,j$ \emph{cofactor}\textsl{\ }%
\label{Cofactor}$A_{ij}$ of $A$ is the scalar $\left( -1\right) ^{i+j}\det
A\left( i|j\right) $, where $A\left( i|j\right) $ denotes the matrix that
you obtain from $A$ by removing the $i$th row and $j$th column.

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

\begin{quotation}
$\left( 
\begin{array}{cc}
a & b \\ 
c & d
\end{array}
\right) $, adjugate: $\left( 
\begin{array}{rr}
d & -b \\ 
-c & a
\end{array}
\right) \medskip $

\emph{Note\quad }The product of a matrix with its adjugate is diagonal, with
the entries on the diagonal equal to the determinant of the matrix. 
\[
\left( 
\begin{array}{cc}
a & b \\ 
c & d
\end{array}
\right) \left( 
\begin{array}{rr}
d & -b \\ 
-c & a
\end{array}
\right) =\,\left( 
\begin{array}{cc}
ad-bc & 0 \\ 
0 & ad-bc
\end{array}
\right) 
\]

\ 
\end{quotation}

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

\begin{quotation}
$\left[ 
\begin{array}{rrrr}
9 & 6 & 7 & -5 \\ 
4 & -8 & -3 & 92 \\ 
-3 & -6 & 7 & 6 \\ 
5 & -5 & 0 & -1
\end{array}
\right] $, adjugate: $\left[ 
\begin{array}{rrrr}
3384 & 469 & -3183 & 7130 \\ 
3329 & 301 & -3200 & -8153 \\ 
4068 & -261 & 6896 & -2976 \\ 
275 & 840 & 85 & -1116
\end{array}
\right] $
\end{quotation}

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

\begin{quotation}
$\det \left[ 
\begin{array}{rrrr}
9 & 6 & 7 & -5 \\ 
4 & -8 & -3 & 92 \\ 
-3 & -6 & 7 & 6 \\ 
5 & -5 & 0 & -1
\end{array}
\right] =\,77531$

\ \vspace{6pt}

$\left[ 
\begin{array}{rrrr}
9 & 6 & 7 & -5 \\ 
4 & -8 & -3 & 92 \\ 
-3 & -6 & 7 & 6 \\ 
5 & -5 & 0 & -1
\end{array}
\right] \left[ 
\begin{array}{rrrr}
3384 & 469 & -3183 & 7130 \\ 
3329 & 301 & -3200 & -8153 \\ 
4068 & -261 & 6896 & -2976 \\ 
275 & 840 & 85 & -1116
\end{array}
\right] $

\ 

$\hspace{1in}=\left[ 
\begin{array}{cccc}
77531 & 0 & 0 & 0 \\ 
0 & 77531 & 0 & 0 \\ 
0 & 0 & 77531 & 0 \\ 
0 & 0 & 0 & 77531
\end{array}
\right] $

\emph{Note\quad }The relationship just demonstrated gives a formula for the
inverse of an invertible matrix $A$. 
\[
A^{-1}=\frac{1}{\det A}\limfunc{adjugate}A 
\]
\end{quotation}

$\strut \,$

\subsection{\ Permanent\label{Permanent}}

The \textsl{permanent} \label{Matrices, permanent}of an $n\times n$ matrix $%
\left( a_{ij}\right) $ is the sum of certain products of the entries.
Specifically, 
\[
\text{permanent}(a_{ij})=\sum_{\sigma }a_{1\sigma \left( 1\right)
}a_{2\sigma \left( 2\right) }\cdots a_{n\sigma \left( n\right) } 
\]
where $\sigma $ ranges over all the permutations of $\left\{ 1,2,\ldots
,n\right\} $. This operation applies to square matrices only.

\ 

\begin{quote}
$\blacktriangleright $ To compute the permanent of a matrix
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point in the matrix.

\item[2.]  From the \textsf{Matrices} submenu, choose \textsf{Permanent}.
\end{enumerate}

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

\begin{quotation}
$\left[ 
\begin{array}{cc}
a & b \\ 
c & d
\end{array}
\right] $, permanent: $ad+bc$

$\left[ 
\begin{array}{ccc}
a_{1,1} & a_{1,2} & a_{1,3} \\ 
a_{2,1} & a_{2,2} & a_{2,3} \\ 
a_{3,1} & a_{3,2} & a_{3,3}
\end{array}
\right] $, permanent: $
\begin{array}{c}
a_{1,1}a_{2,2}a_{3,3}+a_{1,1}a_{2,3}a_{3,2}+a_{2,1}a_{1,2}a_{3,3} \\ 
\quad +a_{2,1}a_{1,3}a_{3,2}+a_{3,1}a_{1,2}a_{2,3}+a_{3,1}a_{1,3}a_{2,2}
\end{array}
$
\end{quotation}

\strut

\subsection{\ Maximum and Minimum Matrix Entries\label%
{Maximum and minimum entries}}

The functions $\max $ and $\min $ applied to a matrix with integer entries
will return the entry with maximum or minimum value.

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

\begin{quotation}
$\max \left[ 
\begin{array}{ccccc}
-85 & -55 & -37 & -35 & 97 \\ 
50 & 79 & 56 & 49 & 63 \\ 
57 & -59 & 45 & -8 & -93
\end{array}
\right] =\allowbreak 97$

\strut

$\min \left[ 
\begin{array}{ccccc}
92 & 43 & -62 & 77 & 66 \\ 
54 & -5 & 99 & -61 & -50 \\ 
-12 & -18 & 31 & -26 & -62 \\ 
1 & -47 & -91 & -47 & -61
\end{array}
\right] =\allowbreak -91$
\end{quotation}

\ 

\subsection{\ Norm\label{Norm}}

Choosing \textsf{Norm }from the \textsf{Matrices} \label{Vector norms (2)}%
submenu gives the $2$-norm of a vector or \label{Matrices, norm}matrix. The 2%
\textsl{-norm} of a vector is the Euclidean length of the vector. 
\[
\left\| 
\begin{array}{c}
a \\ 
b
\end{array}
\right\| =\,\sqrt{a^{2}+b^{2}}\qquad \qquad \left\| 
\begin{array}{c}
a \\ 
b \\ 
c \\ 
d
\end{array}
\right\| =\,\sqrt{a^{2}+b^{2}+c^{2}+d^{2}} 
\]
The \textsl{2-norm} of a matrix $A$ is its largest singular value---the
number defined by 
\[
\left\| A\right\| =\max_{x\neq 0}\frac{\left\| Ax\right\| }{\left\|
x\right\| } 
\]

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

\begin{quotation}
$\left( 
\begin{array}{cc}
2 & 3 \\ 
5 & 7
\end{array}
\right) $, 2-norm: $\frac{1}{2}\sqrt{89}+\frac{1}{2}\sqrt{85}\vspace{8pt}$%
\qquad

$\left( 
\begin{array}{rrr}
2 & -1 & 0 \\ 
-1 & 2 & -1 \\ 
0 & -1 & 2
\end{array}
\right) $, 2-norm: $2+\sqrt{2}$
\end{quotation}

You obtain a numerical approximation if at least one entry is in
floating-point form.

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

\begin{quotation}
$\left( 
\begin{array}{cc}
.2 & .3 \\ 
.5 & .7
\end{array}
\right) $, 2-norm: $.93268$
\end{quotation}

The 2-norm of a matrix can also be obtained with double brackets.

\begin{quote}
$\blacktriangleright $ To put norm symbols around a matrix
\end{quote}

\begin{enumerate}
\item[1.]  Select the matrix by using the mouse or by pressing \textsc{shift
+ right arrow}.

\item[2.]  Click\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{fences.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
24.0625pt;original-height 22.5625pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'FENCES.WMF';file-properties "XNPEU";}}or choose 
\textsf{Insert + Brackets}. Select the norm symbols. Choose \textsf{Ok}.
\end{enumerate}

\subsection{\ Condition Number\label{Condition number}}

The \textsl{condition number} of an invertible \label%
{Matrices, condition number}matrix $A$ is the product of the 2-norm of $A$
and the 2-norm of $A^{-1}$. This number measures the sensitivity of some
solutions of linear equations $Ax=b$ to perturbations in the entries of $A$
and $b$. The matrix with condition number 1 is ``perfectly conditioned.''

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Condition Number}
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{ccc}
0 & 1 & 0 \\ 
0 & 0 & 1 \\ 
1 & 0 & 0
\end{array}
\right] $, condition number: $1$

\ \ 

$\left[ 
\begin{array}{rrr}
5 & -5 & -3 \\ 
-3 & 0 & 5 \\ 
1 & 5 & 4
\end{array}
\right] $, condition number: $2.8249$

\ \ 

$\left[ 
\begin{array}{rrrr}
2 & -1 & 0 & 0 \\ 
-1 & 2 & -1 & 0 \\ 
0 & -1 & 2 & -1 \\ 
0 & 0 & -1 & 2
\end{array}
\right] $, condition number: $\left( \frac{5}{2}+\frac{1}{2}\sqrt{5}\right)
\left( \frac{3}{2}+\frac{1}{2}\sqrt{5}\right) =9.47214$

\ 

$\left[ 
\begin{array}{cccc}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ 
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \frac{1}{5} \\ 
\frac{1}{3} & \frac{1}{4} & \frac{1}{5} & \frac{1}{6} \\ 
\frac{1}{4} & \frac{1}{5} & \frac{1}{6} & \frac{1}{7}
\end{array}
\right] $, condition number: $15514$

\ 

$\left[ 
\begin{array}{cc}
1 & 1 \\ 
1 & 1.00001
\end{array}
\right] $, condition number: $400002$
\end{quotation}

These final two matrices are extremely ill conditioned. Small changes in
some entries of $A$ or $b$ may result in large changes in the solution to
linear equations of the form $Ax=b$ in these two cases.\medskip

\subsection{\ Exponential Functions\label{Exponential function, matrices}}

The most natural way to define $e^{M}$ is to \label%
{Matrices, exponential function} imitate the power series for $e^{x}:$

\begin{itemize}
\item  $e^{x}=\,1+x+\frac{1}{2}x^{2}+\frac{1}{6}x^{3}+\frac{1}{24}%
x^{4}+\cdots $

\item  $e^{M}=\,1+M+\frac{1}{2}M^{2}+\frac{1}{6}M^{3}+\frac{1}{24}%
M^{4}+\cdots $
\end{itemize}

In general, 
\[
e^{tM}=\sum_{k=0}^{\infty }\frac{\left( tM\right) ^{k}}{k!} 
\]
To evaluate the expression $e^{M}$ for a matrix $M$, leave the insertion
point in the expression $e^{M}$ and choose \textsf{Evaluate}, as shown in
the following examples. First define 
\[
A=\left[ 
\begin{array}{cc}
1 & 2 \\ 
0 & 3
\end{array}
\right] \text{, }B=\left[ 
\begin{array}{cc}
1 & 2 \\ 
0 & 1
\end{array}
\right] \text{, }C=\left[ 
\begin{array}{ccc}
0 & 1 & 0 \\ 
0 & 0 & 1 \\ 
0 & 0 & 0
\end{array}
\right] \text{, }D=\left[ 
\begin{array}{ccc}
1 & 3 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array}
\right] 
\]

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

\begin{quotation}
$e^{A}=\,\left[ 
\begin{array}{cc}
\frac{2}{3}e^{-1}+\frac{1}{3}e^{5} & \frac{1}{3}e^{5}-\frac{1}{3}e^{-1} \\ 
\frac{2}{3}e^{5}-\frac{2}{3}e^{-1} & \frac{1}{3}e^{-1}+\frac{2}{3}e^{5}
\end{array}
\right] \vspace{6pt}$

$e^{tA}=\,\left[ 
\begin{array}{cc}
\frac{2}{3}e^{-t}+\frac{1}{3}e^{5t} & \vspace{6pt}\frac{1}{3}e^{5t}-\frac{1}{%
3}e^{-t} \\ 
\frac{2}{3}e^{5t}-\frac{2}{3}e^{-t} & \frac{1}{3}e^{-t}+\frac{2}{3}e^{5t}
\end{array}
\right] \vspace{6pt}$

$e^{A+B}=\,\left[ 
\begin{array}{cc}
e^{2} & 2e^{4}-2e^{2} \\ 
0 & e^{4}
\end{array}
\right] \vspace{6pt}$

$e^{A}e^{B}=\,\left[ 
\begin{array}{cc}
e^{2} & e^{2}+e^{4} \\ 
0 & e^{4}
\end{array}
\right] \vspace{6pt}$

$De^{tC}D^{-1}=\,\,\left[ 
\begin{array}{ccc}
1 & t & \frac{1}{2}t^{2}+3t \\ 
0 & 1 & t \\ 
0 & 0 & 1
\end{array}
\right] \vspace{6pt}$

$e^{DtCD^{-1}}=\,\left[ 
\begin{array}{ccc}
1 & t & \frac{1}{2}t^{2}+3t \\ 
0 & 1 & t \\ 
0 & 0 & 1
\end{array}
\right] \vspace{6pt}$
\end{quotation}

Note that one of the properties of exponents that holds for real numbers
fails for matrices. The equality $e^{A+B}=e^{A}e^{B}$ requires that $AB=BA$,
and this property fails to hold for the matrices in the example. However,
exponentiation preserves the property of similarity, as demonstrated by $%
De^{tC}D^{-1}=e^{DtCD^{-1}}$.\medskip

\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{Polynomials and Vectors Associated with a Matrix}{}{}{%
DM8-6.tex#Characteristic, polynomial}

\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{Matrices}{}{}{%
DMIndex.tex#Matrices}, \hyperref{Matrix operators}{}{}{DMIndex.tex#Matrix
operators}}

\end{document}
