%% 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="31">}
%TCIDATA{<META NAME="GraphicsSave" CONTENT="32">}
%TCIDATA{<META NAME="Title" CONTENT="Matrix Algebra">}
%TCIDATA{Created=Mon Aug 19 14:52:24 1996}
%TCIDATA{LastRevised=Sun Mar 30 18:19:53 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="DM8-2.tex">}
%TCIDATA{<LINK REL="parent" HREF="DM8.tex">}
%TCIDATA{<LINK REL="previous" HREF="DM7-9.tex">}
%TCIDATA{<LINK REL="top" HREF="DM0-0.tex">}
%TCIDATA{MapleDefs=
%$A=\left[ \MATRIX{3,3}{c}\VR{,,c,,,}{,,c,,,}{,,c,,,}{,,c,,,}\HR{,,,}\CELL{-85}\CELL{-55}\CELL{-37}\CELL{-35}\CELL{97}\CELL{50}\CELL{79}\CELL{56}\CELL{49}\right]  $
%}


\input{tcilatex}
\begin{document}


\section{Matrix Algebra\label{Matrix Algebra}}

Matrices are used throughout mathematics and in related fields such as
physics, engineering, economics, and statistics. The algebra of matrices
provides a model for the study of vector spaces and linear transformations.

\subsection{\ Working with Matrices}

A rectangular array of mathematical expressions is called a \textsl{matrix}.
A matrix with $m$ rows and $n$ columns is referred to as an $m\times n\,$
matrix. Matrices are sometimes referred to simply as \textsl{arrays,} and an 
$m\times 1$ or $1\times n$ array is also called a \textsl{vector} 
\index{Vector@Vector}\label{Vector}.

\index{Matrices@Matrices}\label{Matrix entries}Entries in matrices can be
real or complex numbers, or mathematical expressions with real or complex
coefficients. Most of the choices from the \textsf{Matrices} submenu will
operate on both real and complex matrices. The \textsf{QR} and \textsf{SVD}
factorizations discussed later in this chapter assume real matrices.

Matrix entries are identified by their row and column number. The matrix can
be considered as a function on pairs of positive integers and, if the matrix
is given a name, this feature can be used to retrieve the entries.

\begin{example}
Define $A=\left[ 
\begin{array}{ccc}
-85 & -55 & -37 \\ 
-35 & 97 & 50 \\ 
79 & 56 & 49
\end{array}
\right] $ using \textsf{Define + New Definition}. Then applying \textsf{%
Evaluate} yields the following. 
\begin{eqnarray*}
A_{2,3} &=&\allowbreak 50 \\
A_{3,3} &=&\allowbreak 49
\end{eqnarray*}
Note that the subscripted row and column numbers are separated by a
comma.\medskip 
\end{example}

\subsection{\ Changing the Appearance of Matrices\label{Screen appearance}}

You can make choices in the \textsf{View} menu that affect the appearance of
matrices on the screen. \textsf{Helper Lines} and \textsf{Input Boxes} can
be \emph{shown} or \emph{hidden}. The default is to show them to make it
easier to handle entries on the screen. However, matrix helper lines and
input boxes may not appear when you preview or print the document. For this
reason, you will generally want to put brackets 
\index{Matrices@Matrices!brackets@brackets}\label{Matrices, brackets}around
matrices. \textsl{Scientific Notebook} generally presents the result of an
operation on matrices with the same brackets as the original matrices. The
choice of round or square brackets does not affect the mathematical
properties of the matrix. Vertical straight brackets are interpreted as a
mathematical operation (called a determinant) so they should be avoided for
general use.

\begin{quote}
$\blacktriangleright $ To enclose a matrix in brackets
\end{quote}

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

\item[2.]  Click 
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{parens.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
0pt;original-height 0pt;cropleft "0";croptop "1";cropright "1";cropbottom
"0";filename 'Parens.wmf';file-properties "XNPEU";}}
\end{enumerate}

\subsection{\ Creating Matrices\label{Creating Matrices}}

\begin{quote}
$\blacktriangleright $ To create a matrix
\end{quote}

\begin{enumerate}
\item[1.]  Click \FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{matrix.wmf}{%
\special{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Matrix.wmf';file-properties "XNPEU";}} or
choose \textsf{Insert + Matrix.}

\item[2.]  Select the number of rows and columns.

\item[3.]  Click \textsf{OK.}

\item[4.]  Type in the entries.
\end{enumerate}

The entries can be any valid mathematical expression. Both real and complex
numbers are legitimate entries, as well as algebraic expressions.

You can move about in a matrix with the arrow keys, by pressing \textsc{tab}
and \textsc{backtab}, and by using the mouse. Pressing \textsc{spacebar}
will move your insertion point through the mathematics or out of the matrix.

\index{Fill matrix@Fill matrix}\label{Fill matrix}A $2\times 2$ matrix can
be entered from a Windows keyboard with \textsc{ctrl + a}. You can also
create an $m\times n$ matrix using \textsf{Fill Matrix} from the \textsf{%
Matrices} submenu. 
\index{Matrices@Matrices!fill matrix@fill matrix}\label%
{Matrices, fill matrix}

\begin{quote}
$\blacktriangleright $ To create a matrix with \textsf{Fill Matrix}
\end{quote}

\begin{enumerate}
\item  From the \textsf{Matrices} submenu, choose \textsf{Fill Matrix.}

\item  Set the row and column numbers.

\item  Choose one of the menu items from the dialog box.%
\FRAME{dtbpF}{3.5051in}{2.06in}{0pt}{}{}{fillmat2.wmf}{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid_file "F";width
3.5051in;height 2.06in;depth 0pt;original-width 249.9375pt;original-height
146.0625pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'FillMat2.wmf';file-properties "XNPEU";}}

\item  Click \textsf{OK}.
\end{enumerate}

The matrix will appear inside the brackets, filled with the entries you
chose. These choices are discussed in the next few paragraphs.

\subsubsection{Zero\label{Matrices, zero}}

You can get an $m\times n$ matrix of all zeroes, for any $m$ and $n$. 
\index{Matrices@Matrices!zero@zero}

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Fill Matrix + Zero }
\end{quote}

\begin{quotation}
$\left( 
\begin{array}{cc}
0 & 0 \\ 
0 & 0
\end{array}
\right) $
\end{quotation}

\subsubsection{Identity\label{Matrices, identity}}

You can get an $n\times n$ identity matrix for any $n$. 
\index{Matrices@Matrices!identity@identity}You can also specify a nonsquare
matrix and choose identity.\ 

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Fill Matrix + Identity }
\end{quote}

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

$\left[ 
\begin{array}{cccc}
1 & 0 & 0 & 0 \\ 
0 & 1 & 0 & 0 \\ 
0 & 0 & 1 & 0
\end{array}
\right] $\quad \quad \quad $\left[ 
\begin{array}{ccc}
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1 \\ 
0 & 0 & 0
\end{array}
\right] $
\end{quotation}

\strut

This operation produces a (square) identity matrix as large as possible and
fills in remaining rows or columns, if any, with zeroes.

\subsubsection{Random 
\index{Matrices@Matrices!random@random}\label{Matrices, random}}

With this option, you get a matrix filled with random integers between $-99$
and $99$.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Fill Matrix + Random }
\end{quote}

\begin{quotation}
$\left[ 
\begin{array}{rr}
-85 & \;\;55 \\ 
36 & -35
\end{array}
\right] $
\end{quotation}

\subsubsection{Random Matrix\label{Random matrix}}

In addition to \textsf{Fill Matrix}, the \textsf{Matrices} submenu contains
the entry \textsf{Random Matrix}. This choice brings up the following dialog.%
\FRAME{dtbpF}{2.8608in}{2.8816in}{0pt}{}{}{randmat.wmf}{\special{language
"Scientific Word";type "GRAPHIC";display "PICT";valid_file "F";width
2.8608in;height 2.8816in;depth 0pt;original-width 204.75pt;original-height
206.25pt;cropleft "0";croptop "1";cropright "1";cropbottom "0";filename
'RandMat.wmf';file-properties "XNPEU";}}This lets you specify the range for
random (integer) entries as well as specify some special types.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Random Matrix }
\end{quote}

($3\times 3$, \textsf{Antisymmetric}, \textsf{Range of Random Entries: }-10
to 10)\medskip

\begin{quotation}
\[
\left[ 
\begin{array}{ccc}
0 & -4 & 7 \\ 
4 & 0 & 8 \\ 
-7 & -8 & 0
\end{array}
\right] 
\]
\end{quotation}

\subsubsection{Jordan Block}

A \textsl{Jordan } 
\index{Jordan block@Jordan block}\label{Jordan block}\textsl{block } 
\index{Matrices@Matrices!Jordan block@Jordan block}\label%
{Matrices, Jordan block}is a square matrix with the same expression along
the main diagonal, ones on the superdiagonal, and zeroes elsewhere. The
dialog box asks for the number of rows and columns and for an item for the
diagonal. Jordan forms that are built from Jordan blocks are discussed in
more detail later in this chapter.

\begin{quote}
$\blacktriangleright $ \textsf{Matrices + Fill Matrix + Jordan block}
\end{quote}

\begin{quotation}
$2\times 2$ with $\lambda $ on diagonal and $4\times 4$ with $x$ on diagonal 
\[
\left( 
\begin{array}{cc}
\lambda & 1 \\ 
0 & \lambda
\end{array}
\right) \qquad \left( 
\begin{array}{cccc}
x & 1 & 0 & 0 \\ 
0 & x & 1 & 0 \\ 
0 & 0 & x & 1 \\ 
0 & 0 & 0 & x
\end{array}
\right) 
\]
\end{quotation}

\subsubsection{Defined by Function\label{Defined by function}}

To use the \textsf{Defined by Function} option, first define a function $%
f(i,j)$ of two variables. Then use \textsf{Defined by Function} to create
the $m\times n$ matrix with $(i,j)$ entry equal to $f(i,j)$ for $1\leq i\leq
m$ and $1\leq j\leq n$.

\begin{example}
\textsl{Hilbert matrices}\label{Hilbert matrix}

Define $f(i,j)=%
\frac{1}{i+j-1}$.

From the \textsf{Fill Matrix} dialog, choose \textsf{Defined by Function}.

Type $f$ in the box for the function name. 
\index{Hilbert matrix@Hilbert matrix}

Set rows and columns to $2$ or $3.$

Click \textsf{OK}. 
\[
\left[ 
\begin{array}{cc}
1%
\vspace{4pt} & \frac{1}{2} \\ 
\frac{1}{2} & \frac{1}{3}
\end{array}
\right] \qquad \left[ 
\begin{array}{ccc}
1 & \frac{1}{2} & \vspace{6pt}\frac{1}{3} \\ 
\vspace{6pt}\frac{1}{2} & \frac{1}{3} & \frac{1}{4} \\ 
\frac{1}{3} & \frac{1}{4} & \frac{1}{5}
\end{array}
\right] 
\]
\end{example}

\begin{example}
\textsl{Vandermonde matrix}\label{Vandermonde matrix}

Define the function $g(i,j)=x_{i}^{j-1}$.

From the \textsf{Fill Matrix} dialog, choose \textsf{Defined by function}.

Enter $g$ for the function name.

Set rows and columns to $4$. 
\index{Vandermonde matrix@Vandermonde matrix} 
\[
\left[ 
\begin{array}{cccc}
1 & x_{1}%
\vspace{6pt} & x_{1}^{2} & x_{1}^{3} \\ 
1 & x_{2}\vspace{6pt} & x_{2}^{2} & x_{2}^{3} \\ 
1 & x_{3}\vspace{6pt} & x_{3}^{2} & x_{3}^{3} \\ 
1 & x_{4} & x_{4}^{2} & x_{4}^{3}
\end{array}
\right] 
\]
\end{example}

You can use \textsf{Fill Matrix} to create a general matrix with entries
such as $a_{i,j}$.

\begin{example}
\textsl{An ``arbitrary'' }$3\times 3$\textsl{\ matrix}

Define the function $a(i,j)=a_{i,j}$.

From the \textsf{Fill Matrix} dialog, choose \textsf{Defined by function.}

Enter $a$ for the function name.

Set rows and columns to $3$. 
\[
\left[ 
\begin{array}{ccc}
a_{1,1} & a_{1,2} & a_{1,3} \\ 
a_{2,1} & a_{2,2} & a_{2,3} \\ 
a_{3,1} & a_{3,2} & a_{3,3}
\end{array}
\right] 
\]
\end{example}

\ 

Note the comma between subscripts. Without the comma, the subscript $ij$
would be interpreted as a product! You can use the following trick to create
a general matrix up to $9\times 9$ with no commas in the subscripts.

\begin{example}
\textsl{Another form for an ``arbitrary'' }$3\times 3$\textsl{\ matrix}

Define the function $a(i,j)=a_{10i+j}$.

From the \textsf{Fill Matrix} dialog, choose \textsf{Defined by function.}

Enter $a$ for the function name.

Set rows and columns to $3$. 
\[
\left[ 
\begin{array}{ccc}
a_{11} & a_{12} & a_{13} \\ 
a_{21} & a_{22} & a_{23} \\ 
a_{31} & a_{32} & a_{33}
\end{array}
\right] 
\]
\end{example}

\begin{example}
\textsl{Constant matrices}

From the \textsf{Fill Matrix} dialog, choose \textsf{Defined by function.}

Enter $5$ for the function name. 
\[
\left[ 
\begin{array}{cc}
5 & 5 \\ 
5 & 5
\end{array}
\right] 
\]
\end{example}

\subsubsection{Band}

The \textsf{Band} option 
\index{Matrices@Matrices!band@band}\label{Matrices, band} requires 
\index{Band matrix@Band matrix}\label{Band matrix} that you enter a list
such as ``$a,b,c$'' with an odd number of entries. This option creates a
matrix with a ``band'' of entries around the main diagonal, up to the width
of your list, and with zeroes elsewhere. The first entry will be the item in
the middle of your list.

\begin{example}
\textsl{A }$2\times 2$\textsl{\ band matrix}

From the \textsf{Fill Matrix} dialog, choose \textsf{Band}.

Type $a$ in the \textsf{Enter band list }box. 
\[
\left[ 
\begin{array}{cc}
a & 0 \\ 
0 & a
\end{array}
\right] 
\]
\end{example}

\begin{example}
\textsl{Band matrices}

From the \textsf{Fill Matrix} dialog, choose \textsf{Band}.

Type $a,b,c$ in the \textsf{Enter Band List }box.

Set rows to $2$ or $5$, and columns to $2$, $5$, or $8$.%
\vspace{6pt} 
\[
\left[ 
\begin{array}{cc}
b & c \\ 
a & b
\end{array}
\right] \quad \quad \left[ 
\begin{array}{ccccc}
b & c & 0 & 0 & 0 \\ 
a & b & c & 0 & 0 \\ 
0 & a & b & c & 0 \\ 
0 & 0 & a & b & c \\ 
0 & 0 & 0 & a & b
\end{array}
\right] \quad \quad \left[ 
\begin{array}{cccccccc}
b & c & 0 & 0 & 0 & 0 & 0 & 0 \\ 
a & b & c & 0 & 0 & 0 & 0 & 0 \\ 
0 & a & b & c & 0 & 0 & 0 & 0 \\ 
0 & 0 & a & b & c & 0 & 0 & 0 \\ 
0 & 0 & 0 & a & b & c & 0 & 0
\end{array}
\right] 
\]
\end{example}

\begin{example}
Choosing \textsf{Band} and entering the lists ``$0$,'' ``$1$,'' and ``$%
0,\lambda ,1$'' generates a zero matrix, an identity matrix, and a Jordan
block, respectively.\vspace{6pt} 
\[
\left[ 
\begin{array}{ccc}
0 & 0 & 0 \\ 
0 & 0 & 0 \\ 
0 & 0 & 0
\end{array}
\right] \qquad \left[ 
\begin{array}{ccc}
1 & 0 & 0 \\ 
0 & 1 & 0 \\ 
0 & 0 & 1
\end{array}
\right] \qquad \left[ 
\begin{array}{ccc}
\lambda & 1 & 0 \\ 
0 & \lambda & 1 \\ 
0 & 0 & \lambda
\end{array}
\right] 
\]
\medskip
\end{example}

\subsection{\ Revising Matrices\label{Revising Matrices}}

Rows 
\index{Matrices@Matrices!rows@rows}\label{Matrices, delete rows}\label%
{Matrices, delete columns}, columns 
\index{Matrices@Matrices!columns@columns}, or a block of rows or columns can
be deleted from a matrix. 
\index{Revise matrix@Revise matrix}

\begin{quote}
$\blacktriangleright $ To delete (a block of) rows or columns
\end{quote}

\begin{enumerate}
\item[1.]  Select a block of rows or columns by using the mouse.

\item[2.]  Press \textsc{del}.
\end{enumerate}

\begin{quote}
$\blacktriangleright $ To add rows or columns of entries
\end{quote}

\begin{enumerate}
\item[1.]  Select the matrix\label{Matrices, insert rows} \label%
{Matrices, insert columns}by placing the insertion point at the right of the
matrix.

\item[2.]  From the \textsf{Edit} menu, choose \textsf{Insert Rows} or 
\textsf{Insert Columns}.

\item[3.]  Make appropriate choices from the dialog box that appears.\medskip
\end{enumerate}

\begin{quotation}
\emph{Tip\quad }In the special case of a vector represented as an $n\times 1$
or $1\times n$ matrix, you can lengthen the vector by placing the insertion
point in the last input box and pressing \textsc{enter}. You can shorten a
vector by placing the insertion point in the last input box and pressing 
\textsc{backspace}. You can start with a display box, or the input boxes
that appear with the fraction, radical, or bracket buttons, and make the
same kinds of changes.
\end{quotation}

\begin{quote}
$\blacktriangleright $ To change the alignment 
\index{Matrices@Matrices!alignment@alignment}\label{Matrices, alignment}of
entries
\end{quote}

\begin{enumerate}
\item[1.]  Select the matrix using the mouse (or starting with the insertion
point at the left of the matrix, press \textsc{shift + right arrow}).

\item[2.]  Click the \textsf{Properties} button%
\FRAME{itbpF}{0.3009in}{0.3009in}{0.0701in}{}{}{property.wmf}{\special%
{language "Scientific Word";type "GRAPHIC";display "PICT";valid_file
"F";width 0.3009in;height 0.3009in;depth 0.0701in;original-width
17.8125pt;original-height 16.9375pt;cropleft "0";croptop "1";cropright
"1";cropbottom "0";filename 'Property.wmf';file-properties "XNPEU";}}.

\item[3.]  Make appropriate choices from the dialog box that appears.\medskip
\end{enumerate}

You can replace a rectangular block in an existing matrix using \textsf{Fill
Matrix.}

\begin{quote}
$\blacktriangleright $ To change a matrix with \textsf{Fill Matrix\label%
{Changing}}
\end{quote}

\begin{enumerate}
\item[1.]  Select a rectangular portion of the matrix by using the mouse.

\item[2.]  Click the \textsf{Fill Matrix} button, or from the \textsf{%
Matrices} submenu, choose \textsf{Fill Matrix.}

\item[3.]  Choose one of the items from the dialog box.

\item[4.]  Click \textsf{OK}.
\end{enumerate}

The selected region of the matrix is filled with the entries that you
chose.\medskip

\begin{example}
To change the lower-right $2\times 2$ corner of the matrix 
\[
\left[ 
\begin{array}{rrr}
1 & 2 & 3 \\ 
5 & 5 & 4 \\ 
7 & 8 & 9
\end{array}
\right] 
\]
to the zero matrix,

Select the lower-right $2\times 2$ corner of the matrix using the mouse.
From the \textsf{Matrices} submenu, choose \textsf{Fill Matrix. }Choose 
\textsf{Zero}. 
\[
\left[ 
\begin{array}{rrr}
1 & 2 & 3 \\ 
5 & 0 & 0 \\ 
7 & 0 & 0
\end{array}
\right] 
\]
\end{example}

The lower-right corner is \emph{replaced} by the $2\times 2$ zero matrix. No
new matrix is created.\medskip

You can delete a block of entries in a matrix by selecting a rectangular
portion of the matrix with the mouse, and pressing \textsc{del}.\medskip

\begin{example}
To delete the entries in the lower-right $2\times 2$ corner of the matrix 
\[
\left[ 
\begin{array}{rrr}
1 & 2 & 3 \\ 
5 & 5 & 4 \\ 
7 & 8 & 9
\end{array}
\right] 
\]

Select the lower-right $2\times 2$ corner of the matrix using the mouse.

Press \textsc{del} to get 
\[
\left[ 
\begin{array}{rrr}
1 & 2 & 3 \\ 
5 & \Box & \Box \\ 
7 & \Box & \Box
\end{array}
\right] 
\]
\bigskip
\end{example}

\subsection{\ Concatenating Matrices\label{Concatenate}}

You can merge two matrices into one if they have the same number of rows.

\begin{quote}
$\blacktriangleright $ To concatenate two matrices
\end{quote}

\begin{enumerate}
\item[1.]  Place two matrices adjacent to each other.

\item[2.]  Leave the insertion point in one of the matrices.

\item[3.]  From the \textsf{Matrices} submenu, 
\index{Concatenate@Concatenate}choose \textsf{Concatenate}. 
\index{Matrices@Matrices!concatenate@concatenate}\label%
{Matrices, concatenate}
\end{enumerate}

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

\begin{quotation}
$\left( 
\begin{array}{cc}
1 & 2 \\ 
3 & 4
\end{array}
\right) \left( 
\begin{array}{cc}
5 & 6 \\ 
7 & 8
\end{array}
\right) $, concatenate : $\left( 
\begin{array}{cccc}
1 & 2 & 5 & 6 \\ 
3 & 4 & 7 & 8
\end{array}
\right) $

\ 

$\left( 
\begin{array}{cc}
x+1 & 2 \\ 
3y & 4t+2
\end{array}
\right) \left( 
\begin{array}{c}
5+w \\ 
\sqrt{7}z
\end{array}
\right) $, concatenate: 
\[
\left( 
\begin{array}{ccc}
x+1 & 2 & 5+w \\ 
3y & 4t+2 & \sqrt{7}z
\end{array}
\right) 
\]
\end{quotation}

\strut

\subsection{\ Reshaping Lists and Matrices\label{Reshaping}}

A list of expressions entered in mathematics and separated by commas can be
turned into a matrix where the entries of the matrix, reading left to right
and top to bottom, are the entries of the list in the given order.

\begin{quote}
$\blacktriangleright $ To make a matrix from a list
\end{quote}

\begin{enumerate}
\item[1.]  Place the insertion point within the list.

\item[2.]  From the \textsf{Matrices} submenu, choose \textsf{Reshape}. 
\index{Matrices@Matrices!reshape@reshape}\label{Matrices, reshape (2)}

\item[3.]  Specify the number of columns.
\end{enumerate}

The number of rows depends on the length of the list. Extra input boxes at
the end are left blank.

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

\begin{quotation}
$45,21,8,19,0,5,15,6$ to 3 columns: $\left[ 
\begin{array}{ccc}
45 & 21 & 8 \\ 
19 & 0 & 5 \\ 
15 & 6 & \Box
\end{array}
\right] $
\end{quotation}

\strut

A matrix filled with data can be reshaped, with the new matrix corresponding
to the same list as the original data.

\begin{quote}
$\blacktriangleright $ To reshape a matrix
\end{quote}

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

\item[2.]  From the \textsf{Matrices} submenu, choose \textsf{Reshape.}

\item[3.]  Specify the new number of columns.
\end{enumerate}

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

\begin{quotation}
$\left[ 
\begin{array}{rrrr}
-85 & -55 & -37 & -35 \\ 
97 & 50 & 79 & 56
\end{array}
\right] $ to 3 columns: $\left[ 
\begin{array}{rrr}
-85 & -55 & -37 \\ 
-35 & 97 & 50 \\ 
79 & 56 & \Box
\end{array}
\right] $
\end{quotation}

$\strut $

\subsection{\ Related topics}

\begin{itemize}
\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{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{Linear Programming}{}{}{DM12-5.tex#Linear programming}

\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 "0.020828";croptop "1";cropright
"1.020828";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}, 
\hyperref{Random}{}{}{DMIndex.tex#Random}}

\end{document}
