\chapter{Review of Matrix Algebra}
 \chaptermark{Matrix Review}
\label{chap:matrixreview}

This book assumes the reader has had a course in linear algebra (or has
self-studied it, always the better approach).  This appendix is intended
as a review of basic matrix algebra, or a quick treatment for those
lacking this background.

\section{Terminology and Notation}

A {\bf matrix} is a rectangular array of numbers.  A {\bf vector} is a
matrix with only one row (a {\bf row vector} or only one column (a {\bf
column vector}).

The expression, ``the (i,j) element of a matrix,'' will mean its element
in row i, column j.

Please note the following conventions:

\begin{itemize}

\item Capital letters, e.g. $A$ and $X$, will be used to denote matrices and
vectors.  

\item Lower-case letters with subscripts, e.g. $a_{2,15}$ and $x_8$,
will be used to denote their elements.

\item Capital letters with subscripts, e.g. $A_{13}$, will be used to
denote submatrices and subvectors.

\end{itemize}

If $A$ is a {\bf square} matrix, i.e., one with equal numbers n of rows and
columns, then its {\bf diagonal} elements are $a_{ii}$, i = 1,...,n.

A square matrix is called {\bf upper-triangular} if $a_{ij} = 0$
whenever $i > j$, with a corresponding definition for {\bf
lower-triangular} matrices.

The {\bf norm} (or {\bf length}) of an n-element vector $X$ is 

\begin{equation}
\parallel{X} \parallel = \sqrt{\sum_{i=1}^n x_i^2}
\end{equation}

\subsection{Matrix Addition and Multiplication}

\begin{itemize}

\item For two matrices have the same numbers of rows and same numbers of
columns, addition is defined elementwise, e.g.

\begin{equation}
\left (
\begin{array}{cc}
1 & 5 \\
0 & 3 \\
4 & 8 
\end{array}
\right ) +
\left (
\begin{array}{cc}
6 & 2 \\
0 & 1 \\
4 & 0 
\end{array}
\right ) =
\left (
\begin{array}{cc}
7 & 7 \\
0 & 4 \\
8 & 8 
\end{array}
\right ) 
\end{equation}

\item Multiplication of a matrix by a {\bf scalar}, i.e., a number, is also
defined elementwise, e.g.

\begin{equation}
0.4 \left (
\begin{array}{cc}
7 & 7 \\
0 & 4 \\
8 & 8 
\end{array}
\right ) =
\left (
\begin{array}{cc}
2.8 & 2.8 \\
0 & 1.6 \\
3.2 & 3.2 
\end{array}
\right ) 
\end{equation}

\item The {\bf inner product} or {\bf dot product} of equal-length
vectors $X$ and $Y$ is defined to be

\begin{equation}
\sum_{k=1}^n x_k y_k
\end{equation}

\item The product of matrices $A$ and $B$ is defined if the number of rows
of $B$ equals the number of columns of $A$ ($A$ and $B$ are said to be {\bf
conformable}).  In that case, the (i,j) element of the product $C$ is
defined to be

\begin{equation}
c_{ij} = \sum_{k=1}^n a_{ik} b_{kj}
\end{equation}

For instance,

\begin{equation}
\left (
\begin{array}{cc}
7 & 6 \\
0 & 4 \\
8 & 8 
\end{array}
\right )
\left (
\begin{array}{cc}
1 & 6 \\
2 & 4 
\end{array}
\right ) =
\left (
\begin{array}{cc}
19 & 66 \\
8 & 16 \\
24 & 80
\end{array}
\right ) 
\end{equation}

It is helpful to visualize $c_{ij}$ as the inner product of row i of $A$
and column j of $B$, e.g. as shown in bold face here:

\begin{equation}
\left (
\begin{array}{cc}
\mathbf 7 & \mathbf 6 \\
0 & 4 \\
8 & 8 
\end{array}
\right )
\left (
\begin{array}{cc}
\mathbf 1 & 6 \\
\mathbf 2 & 4 
\end{array}
\right ) =
\left (
\begin{array}{cc}
\mathbf 19 & 66 \\
8 & 16 \\
24 & 80
\end{array}
\right ) 
\end{equation}

\item Matrix multiplication is associative and distributive, but in
general not commutative:

\begin{equation}
A(BC) = (AB)C
\end{equation}

\begin{equation}
A(B+C) = AB + AC
\end{equation}

\begin{equation}
AB \neq BA
\end{equation}

\end{itemize}

\section{Matrix Transpose}

\begin{itemize}

\item The transpose of a matrix A, denoted $A'$ or $A^{T}$, is obtained by
exchanging the rows and columns of A, e.g.

\begin{equation}
\left (
\begin{array}{cc}
7 & 70 \\
8 & 16 \\
8 & 80
\end{array}
\right )' =
\left (
\begin{array}{ccc}
7 & 8 & 8 \\
70 & 16 & 80
\end{array}
\right )
\end{equation}

\item If $A + B$ is defined, then

\begin{equation}
(A+B)' = A' + B'
\end{equation}

\item If $A$ and $B$ are conformable, then

\begin{equation}
(AB)' = B'A'
\end{equation}

\end{itemize}

\section{Linear Independence}

Equal-length vectors $X_1$,...,$X_k$ are said to be {\bf linearly
independent} if it is impossible for

\begin{equation}
a_1 X_1 +
... +
a_k X_k = 0
\end{equation}

unless all the $a_i$ are 0.

\section{Determinants}

Let $A$ be an $n \times n$ matrix.  The definition of the determinant of
$A$, det(A), involves an abstract formula featuring permutations.  It will be
omitted here, in favor of the following computational method.

Let $A_{-(i,j)}$ denote the submatrix of $A$ obtained by deleting its
i$^{th}$ row and j$^{th}$ column.  Then the determinant can be computed
recursively across the k$^{th}$ row of $A$ as

\begin{equation}
det(A) =
\sum_{m=1}^n (-1)^{k+m} det(A_{-(k,m)})
\end{equation}

where

\begin{equation}
det
\left (
\begin{array}{cc}
s & t   \\
u & v 
\end{array}
\right ) = sv -tu
\end{equation}

Generally, determinants are mainly of theoretical importance, but they
often can clarify one's understanding of concepts.

\section{Matrix Inverse}
\label{matrixinverse}

\begin{itemize}

\item The {\bf identity} matrix $I$ of size $n$ has 1s in all of its
diagonal elements but 0s in all off-diagonal elements.  It has the
property that $AI = A$ and $IA = A$ whenever those products are defined.

\item The $A$ is a square matrix and $AB = I$, then $B$ is said to be the
{\bf inverse} of $A$, denoted $A^{-1}$.  Then $BA = I$ will hold as well.

\item $A^{-1}$ exists if and only if its rows (or columns) are
linearly independent.

\item $A^{-1}$ exists if and only if $det(A) \neq 0$.

\item If $A$ and $B$ are square, conformable and invertible, then $AB$ is also
invertible, and

\begin{equation}
(AB)^{-1} = B^{-1} A^{-1}
\end{equation}

\end{itemize}

A matrix $U$ is said to be {\bf orthogonal} if its rows each have norm 1
and are orthogonal to each other, i.e., their inner product is 0.  $U$ thus
has the property that $U U' = I$ i.e., $U^{-1} = U$.

The inverse of a triangular matrix is easily obtained by something called
{\bf back substitution}.

Typically one does not compute matrix inverses directly.  A common
alternative is the {\bf QR decomposition}:  For a matrix $A$,
matrices Q and R are calculated so that $A = QR$, where $Q$ is an orthogonal
matrix and $R$ is upper-triangular.  

If $A$ is square and invertible, $A^{-1}$ is easily found: 

\begin{equation}
\label{qr}
A^{-1} = (QR)^{-1} = R^{-1} Q'
\end{equation}

Again, though, in some cases $A$ is part of a more complex system, and the
inverse is not explicitly computed.

\section{Eigenvalues and Eigenvectors}

Let $A$ be a square matrix.\footnote{For nonsquare matrices, the
discussion here would generalize to the topic of {\bf singular value
decomposition}.}  

\begin{itemize}

\item A scalar $\lambda$ and a nonzero vector $X$ that satisfy

\begin{equation}
AX = \lambda X
\end{equation}

are called an {\bf eigenvalue} and {\bf eigenvector} of $A$, respectively.

\item If $A$ is symmetric and real, then it is {\bf diagonalizable},
i.e., there exists an orthogonal matrix $U$ such that

\begin{equation}
\label{diagonalize}
U'AU = D
\end{equation}

for a diagonal matrix $D$.  The elements of $D$ are the eigenvalues of
$A$, and the columns of $U$ are the eigenvectors of $A$.

A different sufficient condition for \ref{diagonalize} is that the
eigenvalues of $A$ are distinct.  In this case, $U$ will not necessarily
be orthogonal.

By the way, this latter sufficient condition shows that ``most'' square
matrices are diagonalizable, if we treat their entries as continous
random variables.  Under such a circumstance, the probability of having
repeated eigenvalues would be 0.

\end{itemize}

\section{Rank of a Matrix}

{\it Definition:}  The rank of a matrix $A$ is the maximal number of
linearly independent columns in $A$.

Let's denote the rank of $A$ by rk($A)$.  Rank has the following
properties:

\begin{itemize}

\item rk($A'$)  = rk($A$)

\item Thus the rank of $A$ is also the maximal number of linearly
independent rows in $A$.

\item Let $A$ be $r \times s$.  Then

\begin{equation}
\textrm{rk}(A) \leq \min(r,s)
\end{equation}

\item rk($A'A$) = rk($A)$

\end{itemize}

\section{Matrix Algebra in R}

The R programming language has extensive facilities for matrix algebra,
introduced here.  Note by the way that R uses column-major order.

A linear algebra vector can be formed as an R vector, or as a one-row or
one-column matrix.

\begin{lstlisting}
> # constructing matrices
> a <- rbind(1:3,10:12)
> a
     [,1] [,2] [,3]
[1,]    1    2    3
[2,]   10   11   12
> b <- matrix(1:9,ncol=3)  
> b
     [,1] [,2] [,3]
[1,]    1    4    7
[2,]    2    5    8
[3,]    3    6    9
# multiplication, etc.
> c <- a %*% b; c + matrix(c(1,-1,0,0,3,8),nrow=2)
     [,1] [,2] [,3]
[1,]   15   32   53
[2,]   67  167  274
> c %*% c(1,5,6)  # note 2 different c's
     [,1]
[1,]  474
# be careful! -- if you extract a submatrix that ends up
# consisting of a single row, the result will be a vector 
# rather than a matrix, unless one specifies drop = FALSE:
> x <- rbind(3:5,c(6,2,9),c(5,12,13))
> class(x[1,])
[1] "numeric"
> x[1,]
[1] 3 4 5
> class(x[1,,drop=FALSE])
[1] "matrix"
> x[1,,drop=FALSE]
     [,1] [,2] [,3]
[1,]    3    4    5
> # transpose, inverse
> t(a)  # transpose
     [,1] [,2]
[1,]    1   10
[2,]    2   11
[3,]    3   12
> u <- matrix(runif(9),nrow=3)
> u
           [,1]       [,2]      [,3]
[1,] 0.08446154 0.86335270 0.6962092
[2,] 0.31174324 0.35352138 0.7310355
[3,] 0.56182226 0.02375487 0.2950227
> uinv <- solve(u)
> uinv
           [,1]      [,2]      [,3]
[1,]  0.5818482 -1.594123  2.576995
[2,]  2.1333965 -2.451237  1.039415
[3,] -1.2798127  3.233115 -1.601586
> u %*% uinv  # note roundoff error
             [,1]          [,2]          [,3]
[1,] 1.000000e+00 -1.680513e-16 -2.283330e-16
[2,] 6.651580e-17  1.000000e+00  4.412703e-17
[3,] 2.287667e-17 -3.539920e-17  1.000000e+00
> # eigenvalues and eigenvectors
> eigen(u)
$values
[1]  1.2456220+0.0000000i -0.2563082+0.2329172i 
-0.2563082-0.2329172i

$vectors
              [,1]                  [,2]                  [,3]
[1,] -0.6901599+0i -0.6537478+0.0000000i 
-0.6537478+0.0000000i
[2,] -0.5874584+0i -0.1989163-0.3827132i 
-0.1989163+0.3827132i
[3,] -0.4225778+0i  0.5666579+0.2558820i  
0.5666579-0.2558820i
> # diagonal matrices (off-diagonals 0)
> diag(3)
     [,1] [,2] [,3]
[1,]    1    0    0
[2,]    0    1    0
[3,]    0    0    1
> diag((c(5,12,13)))
     [,1] [,2] [,3]
[1,]    5    0    0
[2,]    0   12    0
[3,]    0    0   13
\end{lstlisting}

We can obtain matrix inverse using {\bf solve()}, e.g.

\begin{lstlisting}
> m <- rbind(1:2,3:4)
> m
     [,1] [,2]
[1,]    1    2
[2,]    3    4
> minv <- solve(m)
> minv
     [,1] [,2]
[1,] -2.0  1.0
[2,]  1.5 -0.5
> m %*% minv  # should get I back
     [,1]         [,2]
[1,]    1 1.110223e-16
[2,]    0 1.000000e+00
\end{lstlisting}

Note the roundoff error, even with this small matrix.  We can try the QR
method, provided to us in R via {\bf qr()}.  In fact, if we just want
the inverse, {\bf qr.solve()} will compute (\ref{qr}) for us.

We can in principle obtain rank from, for example, the {\bf rank}
component from the output of {\bf qr()}.  Note however that although
rank is clearly defined in theory, the presence of roundoff error in
computation make may rank difficult to determine reliably.