15-00-KroneckerProduct.tex

\documentclass[12pt]{article}
\usepackage{pmmeta}
\pmcanonicalname{KroneckerProduct}
\pmcreated{2013-03-22 13:33:31}
\pmmodified{2013-03-22 13:33:31}
\pmowner{Mathprof}{13753}
\pmmodifier{Mathprof}{13753}
\pmtitle{Kronecker product}
\pmrecord{7}{34163}
\pmprivacy{1}
\pmauthor{Mathprof}{13753}
\pmtype{Definition}
\pmcomment{trigger rebuild}
\pmclassification{msc}{15-00}
\pmsynonym{tensor product (for matrices)}{KroneckerProduct}
\pmsynonym{direct product}{KroneckerProduct}

\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}
\newcommand{\rank}{\mathop{\mathrm{rank}}}
\newcommand{\trace}{\mathop{\mathrm{trace}}}

{\bf Definition.}
Let $A=(a_{ij})$ be a $n\times n$ matrix  and 
let $B$ be a $m\times m$ matrix. Then the
\emph{Kronecker product} of $A$ and $B$ is 
the $mn\times mn$ block matrix
\begin{eqnarray*}
A\otimes B &=&\left( \begin{array}{ccc} 
a_{11} B & \cdots & a_{1n} B \\
\vdots & \ddots & \vdots \\
a_{n1} B & \cdots & a_{nn} B \\
\end{array} \right).
\end{eqnarray*}
The Kronecker product is also known as the \emph{direct product}
or the \emph{tensor product} \cite{eves}. 


{\bf Fundamental properties} \cite{eves, kailath}
\begin{enumerate}
\item The product is bilinear. If $k$ is a scalar, and $A,B$ and $C$ 
are square matrices, such that $B$ and $C$
are of the same order, then
\begin{eqnarray*}
A\otimes (B+C) &=& A\otimes B + A\otimes C,\\ 
(B+C)\otimes A &=& B\otimes A + C\otimes A,\\
k(A\otimes B) &=& (kA)\otimes B = A\otimes (kB).
\end{eqnarray*}
\item If $A,B,C,D$ are square matrices such that the products $AC$ and $BD$ 
exist, then $(A\otimes B)(C\otimes D)$ exists and 
\begin{eqnarray*}
(A\otimes B)(C\otimes D) &=& AC\otimes BD.
\end{eqnarray*}
If $A$ and $B$ are invertible matrices, then
\begin{eqnarray*}
(A\otimes B)^{-1} &=& A^{-1} \otimes B^{-1}.
\end{eqnarray*}
\item If $A$ and $B$ are square matrices, then for the transpose ($A^T$) we have 
\begin{eqnarray*}
(A\otimes B)^{T} &=& A^{T} \otimes B^{T}.
\end{eqnarray*}
\item Let $A$ and $B$ be square matrices of orders $n$ and $m$, respectively. 
If $\{\lambda_i | i=1,\ldots,n \}$ are the eigenvalues of $A$ and 
 $\{\mu_j | j=1,\ldots, m \}$ are the eigenvalues of $B$, then 
$\{\lambda_i \mu_j |  i=1,\ldots, n, \, j=1,\ldots, m \}$ are the eigenvalues of 
$A\otimes B$. Also, 
\begin{eqnarray*}
\det (A\otimes B) &=& (\det A)^m (\det B)^n, \\
\rank (A\otimes B) &=& \rank A\, \rank B, \\
\trace (A\otimes B) &=& \trace A\, \trace B, \\
\end{eqnarray*}
\end{enumerate}

\begin{thebibliography}{9}
\bibitem {eves} H. Eves,
        \emph{Elementary Matrix Theory},
        Dover publications, 1980.
\bibitem {kailath} T. Kailath, A.H. Sayed, B. Hassibi,
        \emph{Linear estimation},
        Prentice Hall, 2000
\end{thebibliography}
%%%%%
%%%%%
\end{document}