\documentclass[12pt]{article} \usepackage[dvipdf]{graphics} \usepackage{amsmath} \setlength{\textwidth}{7.5in} \setlength{\oddsidemargin}{-0.5in} \setlength{\topmargin}{-1in} \setlength{\textheight}{10.5in}%\setlength{\footskip}{-4in} \newtheorem{theorem}{Theorem} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{approximation}[theorem]{Approximation} \newtheorem{definition}[theorem]{Definition} \newenvironment{proof}{{\noindent\bf Proof\ }}{\hfill$\BOX$ \\} \newenvironment{justify}{{\noindent\bf Justification\ }}{$\BOX$ \\} \newtheorem{conjecture}[theorem]{Conjecture} \newcommand{\BC}{\begin{center}} \newcommand{\EC}{\end{center}} \newcommand{\BE}{\begin{equation}} \newcommand{\EE}{\end{equation}} \newcommand{\BEA}{\begin{eqnarray}} \newcommand{\EEA}{\end{eqnarray}} \newcommand{\BEAS}{\begin{eqnarray*}} \newcommand{\EEAS}{\end{eqnarray*}} \newcommand{\BEN}{\begin{enumerate}} \newcommand{\EEN}{\end{enumerate}} \newcommand{\BEG}{\begin{example}\rm} \newcommand{\EEG}{\end{example}} \newcommand{\BTHM}{\begin{theorem}} \newcommand{\ETHM}{\end{theorem}} \newcommand{\BCOR}{\begin{corollary}} \newcommand{\ECOR}{\end{corollary}} \newcommand{\BDEF}{\begin{definition}} \newcommand{\EDEF}{\end{definition}} \newcommand{\BPF}{\begin{proof}\rm} \newcommand{\EPF}{\end{proof}} \newcommand{\BRMK}{\begin{remark}\rm} \newcommand{\ERMK}{\end{remark}} \newcommand{\BLEM}{\begin{lemma}} \newcommand{\ELEM}{\end{lemma}} \newcommand{\BCON}{\begin{conjecture}} \newcommand{\ECON}{\end{conjecture}} \newcommand{\BAPR}{\begin{approximation}} \newcommand{\EAPR}{\end{approximation}} \newcommand{\BPROP}{\begin{proposition}} \newcommand{\EPROP}{\end{proposition}} \newcommand{\bthm}{\begin{theorem}} \newcommand{\ethm}{\end{theorem}} \newcommand{\hs}[1]{\hspace{#1}} \newcommand{\vs}[1]{\vspace{#1}} \newcommand{\ra}{\rightarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\natlog}{\ell n} \newcommand{\E}{\mbox{E}} % expectation operator \newcommand{\Var}{\mbox{Var}} % Var operator \newcommand{\Bias}{\mbox{Bias}} % Bias operator \newcommand{\MSE}{\mbox{MSE}} % MSE operator \newcommand{\Cov}{\mbox{Cov}} % Cov operator \newcommand{\Corr}{\mbox{Corr}} % Corr operator \newcommand{\sig}{\mbox{$\sigma^2$}} \newcommand{\sigsqn}{\mbox{$\sigma_n^2$}} \newcommand{\norm} [1]{\mbox{$\parallel{#1}\parallel$}} \newcommand{\floor}[1]{\mbox{$\lfloor #1 \rfloor$}} \newcommand{\cid}{\mbox{$ \ \stackrel{\cal D}{\rightarrow} \ $}} \newcommand{\summ}[1]{\sum_{#1=1}^{n}} \newcommand{\BOX}{\rule[-2pt]{4pt}{8pt}} \newcommand{\itbold}[1]{\mbox{\boldmath $#1$}} \newcommand{\zee}{\hbox{{\sf Z}\kern-.3550em\hbox{{\sf Z}} }} \newcommand{\real}{\hbox{{I\kern-.1667em\hbox{R}}}} \newcommand{\nn}{{\rm I\hspace{-0.7mm}N}} \newcommand{\mm}{{\rm I\hspace{-0.7mm}M}} \newcommand{\ee}{{\rm I\hspace{-0.7mm}E}} \newcommand{\ff}{{\rm I\hspace{-0.7mm}F}} \newcommand{\ind}{{\rm 1\hspace{-.9mm}l}} \newcommand{\rr}{{\rm I\hspace{-0.7mm}R}} \newcommand{\zz}{{\rm Z\hspace{-2mm}Z}} \newcommand{\qq}{{\rm I\hspace{-1.8mm}Q}} \newcommand{\nin}{\,\,{\rm /\hspace{-3.4mm}\in}\,\,} \begin{document} \noindent Matt Jones \hfill Statistical Methods II\hfill 3260-L14\\ \begin{center} {\bf \Large Review of Matrices} \end{center} \section*{Basics} An $m\times n$ matrix is a set of numbers arranged in an array of $m$ rows and $n$ columns:\[A_{m \times n} = \left(% \begin{array}{cccc} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2} & \cdots & a_{m n} \\ \end{array}% \right)\] \noindent The dimension of the matrix $A$ is $m\times n$, and is often displayed underneath the matrix name as a reminder. A matrix is square if its \# of rows = its \# of columns (if $m=n$). \BEG \EEG \vspace{4cm} \noindent A column vector is a matrix with only one column. A row vector is a matrix with only one row. \BEG \EEG \vspace{4cm} \noindent The transpose $A^T$ of the matrix $A$ is obtained by interchanging the rows and columns of $A$.\BEG \EEG \vspace{6cm} \noindent Note the transpose of an $m\times n$ matrix has dimension $n \times m$.\\ \noindent Two matrices $A$ and $B$ are equal iff they have the same dimension and their corresponding elements are equal: $a_{ij} = b_{ij}$ for all $i,j$. \BEG \EEG \vspace{4cm} \subsection*{Addition and Subtraction} We can add or subtract matrices with the same dimensions. For two $m\times n$ matrices $A$ and $B$, \[A_{m \times n} + B_{m\times n} = \left(% \begin{array}{cccc} a_{11}+b_{11} & a_{12} +b_{12}& \cdots & a_{1n}+b_{1n} \\ a_{21} + b_{21}& a_{22}+b_{22} & \cdots & a_{2n} +b_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} + b_{m 1} & a_{m 2}+b_{m 2}& \cdots & a_{m n}+b_{m n} \\ \end{array}% \right)\] Similarly, \[A_{m \times n} - B_{m\times n} = \left(% \begin{array}{cccc} a_{11}-b_{11} & a_{12} -b_{12}& \cdots & a_{1n}-b_{1n} \\ a_{21} - b_{21}& a_{22}-b_{22} & \cdots & a_{2n} -b_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} - b_{m 1} & a_{m 2}-b_{m 2}& \cdots & a_{m n}-b_{m n} \\ \end{array}% \right)\] \BEG \EEG \vspace{4cm} \section*{Multiplication by a Scalar} A scalar is a number. Given the $m\times n$ matrix $A$ \[A = \left(% \begin{array}{cccc} a_{11} & a_{12}& \cdots & a_{1n}\\ a_{21} & a_{22} & \cdots & a_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ a_{m 1} & a_{m 2}& \cdots & a_{m n}\\ \end{array}% \right)\]and a scalar $k$, the product $kA$ is defined to be \[kA = \left(% \begin{array}{cccc} ka_{11} & ka_{12}& \cdots & ka_{1n}\\ ka_{21} & ka_{22} & \cdots & ka_{2n}\\ \vdots & \vdots & \ddots & \vdots \\ ka_{m 1} & ka_{m 2}& \cdots & ka_{m n}\\ \end{array}% \right)\] \BEG \EEG \vspace{4cm} \section*{Multiplication of Matrices} Given matrices $A_{m\times p}$ and $B_{p\times n}$, their product $C_{m\times n} = A_{m\times p} B_{p\times n}$ is given by \[C_{m\times n} = A_{m\times p} B_{p\times n} = \left(% \begin{array}{cccc} \sum_{k=1}^pa_{1k}b_{k1} & \sum_{k=1}^pa_{1k}b_{k2} & \cdots & \sum_{k=1}^pa_{1k}b_{kn}\\ \sum_{k=1}^pa_{2k}b_{k1} & \sum_{k=1}^pa_{2k}b_{k2} & \cdots & \sum_{k=1}^pa_{2k}b_{kn}\\ \vdots & \vdots & \ddots & \vdots \\ \sum_{k=1}^pa_{mk}b_{k1} & \sum_{k=1}^pa_{mk}b_{k2}& \cdots & \sum_{k=1}^pa_{mk}b_{kn}\\ \end{array}% \right)\] \BEG \EEG \vspace{6cm} \noindent When obtaining the product $AB$, we say ``$B$ is pre-multiplied by $A$", or ``$A$ is post-multiplied by $B$". Note the dimension of $A_{m\times p} B_{p\times n}$ is $m \times n$. \section*{Special Matrix Types} Matrix $A$ is \underline{symmetric} if $A = A^T$. That is, $A$ is symmetric iff $a_{ij} = a_{ji}$ for each $i,j \in\{1,\ldots n\}$, and $n=$ \# of columns or rows. Note that all symmetric matrices are square.\BEG \EEG \vspace{4cm} \noindent A \underline{diagonal matrix} is a square matrix whose only non-zero elements are along the diagonal.\BEG \EEG \vspace{4cm} \noindent The \underline{identity matrix} $I$ is a diagonal matrix whose diagonal elements are 1s.\BEG \EEG \vspace{4cm} \noindent Pre- or post-multiplying a matrix $A$ by $I$ (with appropriate dimension) yields $A$:\[I_{m\times m} A_{m\times n} = A_{m\times n}I_{n\times n} = A_{m \times n}\] A \underline{scalar matrix} is a diagonal matrix whose main diagonal elements are the same, and can be expressed as the product of a scalar and $I$: \BEG \EEG \vspace{5cm} \noindent We use the notation $1_{m\times 1}$, $J_{m\times m}$, and $0_{m\times 1}$ to mean \[1_{m\times 1} = \left(% \begin{array}{c} 1 \\ 1 \\ \vdots \\ 1 \\ \end{array}% \right), \qquad J_{m\times m} = \left(% \begin{array}{cccc} 1 & 1 & \cdots & 1 \\ 1 & 1 & \cdots & 1 \\ \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & \cdots & 1 \\ \end{array}% \right),\qquad 0_{m\times 1} = \left(% \begin{array}{c} 0 \\ 0 \\ \vdots \\ 0 \\ \end{array}% \right)\] \section*{Linear Dependence} The columns of a matrix are \underline{linearly dependent} if one of them is a linear combination of the others. Think of the columns in a matrix as vectors $C_1, C_2, \ldots, C_n$. When $n$ scalars $k_1, \ldots, k_n$, not all zero, can be found such that \BEA \label{lindep}\sum_{i=1}^n k_iC_i = 0,\EEA the $n$ column vectors are linearly dependent. If the only solution to (\ref{lindep}) is $k_i = 0 \forall i$, the $n$ column vectors are \underline{linearly independent}.\BEG \EEG \vspace{7cm} \noindent The rank of a matrix is the maximum number of linearly independent columns. For matrices $A$ and $B$ above, rank($A$) = 3 and rank($B$) = 3. Rank is equivalently defined as the maximum number of linearly independent rows. \section*{Inverse of a Matrix} The \underline{inverse} of a square matrix $A$ is denoted $A^{-1}$ and satisfies \[A A^{-1} = A^{-1} A = I\] when it exists. \BEG \EEG \vspace{7cm} \section*{Basic Rules for Matrices} \noindent For the scalar $k$ and matrices $A, B, C$ of appropriate dimension, \BEAS A+B &=& B+A \qquad \qquad \qquad \;\;\;\;\,\,\,\textrm{(commutative property of matrix addition)}\\ (A+B)+C &=& A+(B+C) \;\;\;\qquad \qquad \textrm{(associative property of matrix addition)}\\ (AB)C &=& A(BC) \qquad \qquad \qquad \;\;\;\;\,\textrm{(associative property of matrix multiplication)}\\ C(A+B)&=& CA + CB \qquad \qquad \qquad \textrm{(distributive law)}\\ k(A+B) &+& kA + k B \qquad \qquad \qquad \,\,\,\textrm{(follows from the distributive law)}\\ (A^T)^T &=& A \\ (A+B)^T &=& A^T + B^T\\ (AB)^T &=& B^TA^T\\ (ABC)^T &=& C^TB^TA^T\\ (AB)^{-1} &=& B^{-1}A^{-1} \qquad \qquad \qquad \;\;\;\,\textrm{(when the inverses exist)}\\ (ABC)^{-1} &=& C^{-1}B^{-1}A^{-1} \qquad \qquad \;\;\;\,\,\, \textrm{(when the inverses exist)}\\ (A^{-1})^{-1} &=& A \qquad\qquad \qquad \qquad \;\;\;\;\,\,\,\, \textrm{(when $A^{-1}$ exists)}\\ (A^T)^{-1} &=& (A^{-1})^T \qquad \qquad \qquad \;\;\;\,\,\,\, \textrm{(when $A^{-1}$ exists)}\EEAS \noindent{\bf Homework: Do the following problems. \begin{enumerate} \item Given the matrices\[A = \left( \begin{array}{cc} 1 & 4 \\ 2 & 6 \\ 3 & -1 \\ \end{array} \right) ,\qquad B = \left( \begin{array}{cc} 2 & 2 \\ 3 & 2 \\ 1 & 7 \\ \end{array} \right) ,\qquad C = \left( \begin{array}{ccc} 3 & 7 & 1 \\ 4 & 7 & 5 \\ \end{array} \right) ,\] obtain $A+B$, $A-B$, $AC$, $AB^{T}$, $B^TA$. \item Prove that for any $2\times 2$ matrix $A$ \[A = \left( \begin{array}{cc} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{array} \right) \] that $A^{-1}$ (if it exists) is given by \[A{-1} = \left( \begin{array}{cc} a_{22}/D & -a_{12}/D \\ -a_{21}/D & a_{11}/D \\ \end{array} \right),\] where $D = a_{11}a_{22} - a_{12}a_{21}$. \item Given the matrices $X$ and $Y$ \[X = \left( \begin{array}{cc} 1 & 2 \\ 1 & 3 \\ 1 & 5 \\ 1 & 6 \\ 1 & 4 \\ \end{array} \right) ,\qquad Y = \left( \begin{array}{c} 6 \\ 9 \\ 23 \\ 25 \\ 17 \\ \end{array} \right) \] evaluate the expression $\beta = (X^{T}X)^{-1}X^TY$. What is the dimension of $\beta$? \end{enumerate}} \end{document}