Kronecker product
KroneckerProduct
2003-04-06 16:27:26
2006-08-08 14:00:38
Definition
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{\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}