Hadamard product
HadamardProduct
2004-03-14 14:38:45
2005-10-28 13:41:27
Definition
Oppenheim inequality
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{\bf Definition}
Suppose $A=(a_{ij})$ and $B=(b_{ij})$ are two $n\times m$-matrices
with entries in some field. Then their \emph{Hadamard product} is
the entry-wise product of $A$ and $B$, that is,
the $n\times m$-matrix $A\circ B$ whose $(i,j)$th entry is $a_{ij} b_{ij}$.
\subsubsection*{Properties}
Suppose $A,B,C$ are matrices of the same size and $\lambda$
is a scalar. Then
\begin{eqnarray*}
A\circ B &=& B\circ A, \\
A\circ (B+C) &=& A\circ B + A\circ C, \\
A\circ (\lambda B) &=& \lambda (A\circ B),
\end{eqnarray*}
\begin{itemize}
\item If $A,B$ are diagonal matrices, then $A\circ B=AB$.
\item (\emph{Oppenheim inequality}) \cite{prasolov}: If $A,B$ are positive definite matrices, and
$(a_{ii})$ are the diagonal entries of $A$, then
$$\det A\circ B \ge \det B\,\prod{a_{ii}}$$
with equality if and only if $A$ is a diagonal matrix.
\end{itemize}
\subsubsection*{Remark}
There is also a Hadamard product for two power series: Then the
Hadamard product of $\sum_{i=1}^\infty a_i$ and $\sum_{i=1}^\infty b_i$ is
$\sum_{i=1}^\infty a_i b_i$.
\begin{thebibliography}{9}
\bibitem{horn} R. A. Horn, C. R. Johnson,
\emph{Topics in Matrix Analysis},
Cambridge University Press, 1994.
\bibitem{prasolov} V.V. Prasolov,
\emph{Problems and Theorems in Linear Algebra},
American Mathematical Society, 1994.
\bibitem{mond} B. Mond, J. E. Pecaric,
\emph{Inequalities for the Hadamard product of matrices},
SIAM Journal on Matrix Analysis and Applications,
Vol. 19, Nr. 1, pp. 66-70.
\PMlinkexternal{(link)}{http://epubs.siam.org/sam-bin/dbq/article/30295}
\end{thebibliography}