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15-00-PartitionedMatrix.tex
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15-00-PartitionedMatrix.tex
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\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{PartitionedMatrix}
\pmcreated{2013-03-22 13:32:55}
\pmmodified{2013-03-22 13:32:55}
\pmowner{mathcam}{2727}
\pmmodifier{mathcam}{2727}
\pmtitle{partitioned matrix}
\pmrecord{11}{34150}
\pmprivacy{1}
\pmauthor{mathcam}{2727}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-00}
%\pmkeywords{Jordan canonical form}
%\pmkeywords{rational canonical form}
%\pmkeywords{smith normal form}
\pmrelated{JordanCanonicalForm}
\pmrelated{JordanCanonicalFormTheorem}
\pmdefines{block matrix}
\pmdefines{sub-matrix}
\pmdefines{submatrix}
\endmetadata
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%%%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\begin{document}
A \emph{partitioned matrix}, or a \emph{block matrix}, is
a matrix $M$ that has been constructed from other smaller matrices.
These smaller matrices are called \emph{blocks} or \emph{sub-matrices}
of $M$.
For instance, if we partition the below $5\times 5$ matrix
as follows
\begin{eqnarray*}
L&=&\left( \begin{array}{cc|ccc}
1 & 0 & 1 & 2 & 3 \\
0 & 1 & 1 & 2 & 3 \\
\hline
2 & 3 & 9 & 9 & 9 \\
2 & 3 & 9 & 9 & 9 \\
2 & 3 & 9 & 9 & 9 \\
\end{array} \right),
\end{eqnarray*}
then we can define the matrices
\begin{equation*}
A=\left( \begin{array}{cc}
1 & 0 \\
0 & 1
\end{array} \right),
B=\left( \begin{array}{ccc}
1 & 2 & 3\\
1 & 2 & 3
\end{array} \right),
C=\left( \begin{array}{cc}
2 & 3 \\
2 & 3 \\
2 & 3
\end{array} \right),
D=\left( \begin{array}{ccc}
9 & 9 & 9 \\
9 & 9 & 9 \\
9 & 9 & 9 \\
\end{array} \right)
\end{equation*}
and write $L$ as
\begin{equation*}
L=\left( \begin{array}{cc}
A & B \\
C & D
\end{array} \right),\, \mbox{or\,\,}
L=\left( \begin{array}{c|c}
A & B \\
\hline
C & D
\end{array} \right).
\end{equation*}
If $A_1,\ldots, A_n$ are square matrices (of possibly
different sizes), then we define the \emph{direct sum} of
the matrices $A_1,\ldots, A_n$
as the partitioned matrix
$$\operatorname{diag}(A_1,\ldots, A_n) =\left( \begin{array}{c|c|c}
A_1 & & \\
\hline
& \ddots & \\
\hline
& & A_n \\
\end{array} \right),$$
where the off-diagonal blocks are zero.
If $A$ and $B$ are matrices of the same size partitioned into blocks of the same size, the partition of the sum is the sum of the partitions.
If $A$ and $B$ are $m\times n$ and $n\times k$ matrices, respectively, then if the blocks of $A$ and $B$ are of the correct size to be multiplied, then the blocks of the product are the products of the blocks.
%%%%%
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\end{document}