\hfill \thepage} %} \newtheorem{acknowledgement}[theorem]{Acknowledgement} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{case}[theorem]{Case} \newtheorem{claim}[theorem]{Claim} \newtheorem{conclusion}[theorem]{Conclusion} \newtheorem{condition}[theorem]{Condition} \newtheorem{criterion}[theorem]{Criterion} \newtheorem{notation}[theorem]{Notation} \newtheorem{problem}[theorem]{Problem} \newtheorem{solution}[theorem]{Solution} \newtheorem{summary}[theorem]{Summary} \newenvironment{proof}[1][P roof]{\textbf{#1.} }{\ \rule{0.5em}{0.5em}} \input{tcilatex} \begin{document} \section{\protect\vspace{1pt}Ma 116 Lecture 4/19/00} \vspace{1pt} \section{Matrices} \subsection{The \protect\vspace{1pt}Definition of a Matrix} An $m\times n$ \emph{matrix} is a rectangular array of quantities arranged in $m$ rows and $n$ columns. (We say the matrix is of order $m$ by $n$). \vspace{1pt} Notation: Let $a_{ij}$ $\ 1\leq i\leq m\qquad 1\leq j\leq n$ be $mn$ quantities. Then the matrix associated with these $a_{ij}$'s is denoted by \vspace{1pt} \begin{center} $A=\left[ a_{ij}\right] _{m\times n}=\left[ \begin{array}{rrrrrrr} a_{11} & a_{12} & . & . & . & . & a_{1n} \\ a_{21} & a_{22} & . & . & . & . & a_{2n} \\ a_{31} & a_{32} & a_{33} & . & . & . & a_{3n} \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ a_{m1} & a_{m2} & . & . & . & . & a_{mn}% \end{array} \right] $ \end{center} The quantities $a_{ij}$ are called the \emph{elements} of the matrix $A$. Definition. Two matrices $A=\left[ a_{ij}\right] $ and $B=\left[ b_{ij}% \right] $ are said to be \emph{equal} $\Leftrightarrow $\ they contain the same number of rows and columns and $a_{ij}=b_{ij}$ \ (\textit{for all)} $% i,j $. \vspace{1pt} \subsection{Special Matrices} \vspace{1pt}There are some special matrices which should be introduced. \vspace{1pt} If $n=1$ $\Longrightarrow A=\left[ a_{n1}\right] _{m\times 1}=\left[ \begin{array}{l} a_{11} \\ a_{12} \\ . \\ . \\ a_{m1}% \end{array} \right] .\qquad $ This is called a column matrix. If $m=1\Longrightarrow A=\left[ a_{1n}\right] _{1\times n}=$ $\left[ a_{11},a_{12},...,a_{1n}\right] .$ This is called a row matrix. \vspace{1pt} Both column and row matrices are referred to as \emph{vectors}. \vspace{1pt} \vspace{1pt}When $m=n,$ the we have a square matrix \ $\left[ \begin{array}{llll} a_{11} & . & . & a_{1n} \\ . & . & . & . \\ . & . & . & . \\ a_{n1} & . & . & a_{nn}% \end{array} \right] $ \vspace{1pt} The matrix $A$ with every element zero is called the \emph{zero} matrix. We will write $A=0$. The identity matrix may be defined as follows: let \ $\delta _{ij}=\left\{ \begin{array}{c} 0\qquad i\neq j \\ 1\text{ \ \ \ \ \ }i=j% \end{array} \right. $ The square matrix $I=\left[ \delta _{ij}\right] _{n\times n}$ is known as the \emph{identity} matrix. \vspace{1pt} \begin{center} \vspace{1pt}% \begin{equation*} I=\left[ \begin{array}{llll} 1 & 0 & . & 0 \\ 0 & 1 & 0 & 0 \\ . & & & 0 \\ 0 & 0 & . & 1% \end{array}% \right] \end{equation*} \vspace{1pt} \end{center} Thus the identity matrix is the matrix with $1^{\prime }s$ along its diagonal and $0^{\prime }s$ everywhere else. \vspace{1pt} \subsection{Operations on Matrices} \vspace{1pt} \paragraph{Addition:} \vspace{1pt} Let $A=\left[ a_{ij}\right] _{m\times n}$ and $B=\left[ b_{ij}\right] _{m\times n}.$ Then \begin{center} \qquad \qquad\ \ \ \begin{equation*} A+B=\left[ a_{ij}+b_{ij}\right] _{m\times n} \end{equation*} \end{center} \vspace{1pt} Thus $A+B$ is a matrix of order $m\times n$ whose $i,j$ entry is $% a_{ij}+b_{ij}$. \vspace{1pt} \paragraph{Example:} $A=\left[ \begin{array}{lll} 1 & -2 & 3 \\ 0 & -1 & 6% \end{array} \right] \qquad B=\left[ \begin{array}{lll} 6 & 4 & 7 \\ -1 & -2 & -6% \end{array} \right] $ \vspace{1pt} \qquad \qquad \begin{equation*} A+B=\left[ \begin{array}{lll} 1+6 & -2+4 & 3+7 \\ 0-1 & -1-2 & 6-6% \end{array}% \right] =\left[ \begin{array}{lll} 7 & 2 & 10 \\ -1 & -3 & 0% \end{array}% \right] \end{equation*} \paragraph{Example:} We can use SNB to add matrices. Thus \vspace{1pt} $\left[ \begin{array}{lll} 1 & -2 & 3 \\ 0 & -1 & 6% \end{array} \right] +\left[ \begin{array}{lll} 6 & 4 & 7 \\ -1 & -2 & -6% \end{array} \right] =\allowbreak \left[ \begin{array}{ccc} 7 & 2 & 10 \\ -1 & -3 & 0% \end{array} \right] $ \vspace{1pt} \subsubsection{Subtraction:} $\vspace{1pt}$ Let $A=\left[ a_{ij}\right] _{m\times n}$ and $B=\left[ b_{ij}\right] _{m\times n}.$ Then $A-B$ is the matrix defined by \qquad \qquad\ \ \ \ \ \ \ \ \ \begin{center} $A-B=\left[ a_{ij}-b_{ij}\right] _{m\times n}$ \end{center} \vspace{1pt} \vspace{1pt}Note: One can add and subtract only matrices of the same order. Such matrices are called \emph{conformable}. \vspace{1pt} \subsection{Scalar Multiplication} \vspace{1pt} Let $k$ be a scalar and $A$ a matrix of real numbers of order $m\times n$. Then $\vspace{1pt}$ \begin{center} \begin{equation*} kA=\left[ k\cdot a_{ij}\right] _{m\times n} \end{equation*} \end{center} \vspace{1pt} Example: \begin{equation*} 5\left[ \begin{array}{cccc} -1 & 0 & 5 & 7 \\ 2 & -8 & 4 & 22 \\ -7 & 1 & 0 & 6 \\ 8 & 3 & -3 & 4% \end{array}% \right] =\allowbreak \left[ \begin{array}{cccc} -5 & 0 & 25 & 35 \\ 10 & -40 & 20 & 110 \\ -35 & 5 & 0 & 30 \\ 40 & 15 & -15 & 20% \end{array}% \right] \end{equation*} \subsection{Some Properties of Addition and Scalar Multiplication} We now list the basic properties of vector addition and scalar multiplication. \paragraph{Theorem} Let $A,$ $B$ and $C$ be conformable $m\times n$ matrices whose entries are real numbers, and $k$ and $p$ arbitrary scalars. Then $1.$ $\ A+B=B+A$. $2.$ \ $A+\left( B+C\right) =\left( A+B\right) +C$ $3.$ \ There is an $m\times n$ matrix $0$ such that $0+A=A$ for each $A.$ $4.$ \ For each $A$ there is an $m\times n$ matrix $-A$ such that $A+\left( -A\right) =0.$ $5.$ \ $k\left( A+B\right) =kA+kB$ $6.$ \ $\left( k+p\right) A=kA+pA$ $7.$ \ $\left( kp\right) A=k\left( pA\right) .$ \paragraph{Proof:} $\left( 1\right) $ $\ A+B=\left[ a_{ij}\right] _{m\times n}+\left[ b_{ij}% \right] _{m\times n}=\left[ a_{ij}+b_{ij}\right] _{m\times n}$ \qquad \qquad $\qquad \qquad \qquad $ $\qquad \qquad \qquad \qquad \qquad =\left[ b_{ij}+a_{ij}\right] _{m\times n} $ \qquad \qquad \qquad commutativity of real numbers. \vspace{1pt} \qquad \qquad \qquad $\qquad \qquad =B+A$ $\left( 4\right) $ \ Note that $\left( -1\right) A=\left[ -a_{ij}\right] _{m\times n}\qquad \Longrightarrow A+\left( -1\right) A=0_{m\times n}$ \vspace{1pt} Remark: We denote $\left( -1\right) A$ by $-A$. \vspace{1pt} \subsection{The Transpose of a Matrix} If $A$ is an $m\times n$ matrix, the \emph{transpose} of $A$, denoted $A^{T}$% , is the $n\times m$ matrix whose entry $a_{st}$ is the same as the entry $% a_{ts}$ in the matrix $A$. Thus one gets the transpose of $A$ by interchanging the rows and the columns of $A.$ \vspace{1pt} \paragraph{Example:} $\left[ \begin{array}{ccc} 1 & 0 & -1 \\ 2 & 3 & -2 \\ 4 & 10 & 9% \end{array} \right] ^{T}=\allowbreak \left[ \begin{array}{ccc} 1 & 2 & 4 \\ 0 & 3 & 10 \\ -1 & -2 & 9% \end{array} \right] $ We note the following: \begin{itemize} \item $\left( A^{T}\right) ^{T}=A$. \item $\left( A+B\right) ^{T}=A^{T}+B^{T}$. \item For any scalar $r$, $\left( rA\right) ^{T}=rA^{T}$. \item If $A$ is a diagonal matrix, then $A=A^{T}$. \item A square matrix is said to be \emph{symmetric} if $A^{T}=A$ and \emph{% skew-symmetric} if $A^{T}=-A$. \end{itemize} \paragraph{\protect\vspace{1pt}Example:} $\left[ \begin{array}{ccc} 1 & 2 & 3 \\ 2 & -1 & 4 \\ 3 & 4 & 0% \end{array} \right] $ is symmetric and $\left[ \begin{array}{ccc} 0 & 2 & 3 \\ -2 & 0 & 4 \\ -3 & -4 & 0% \end{array} \right] $ is skew-symmetric. \subsubsection{\protect\vspace{1pt}} \end{document} %%%%%%%%%%%%%%%%%%%% End /document/lec_4_19_00.tex %%%%%%%%%%%%%%%%%%%%