Hadamard product

Definition Suppose $A=(a_{ij})$ and $B=(b_{ij})$ are two $n\times m$ -matrices with entries in some field. Then their 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 $a_{ij}b_{ij}$ .

Properties

Suppose $A, B, C$ are matrices of the same size and $\lambda$ is a scalar. Then

$\displaystyle A\circ B$	$\displaystyle=$	$\displaystyle B\circ A,$
$\displaystyle A\circ(B+C)$	$\displaystyle=$	$\displaystyle A\circ B+A\circ C,$
$\displaystyle A\circ(\lambda B)$	$\displaystyle=$	$\displaystyle\lambda(A\circ B),$

•

If $A, B$ are diagonal matrices, then $A\circ B=AB$ .
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(Oppenheim inequality) [2]: If $A, B$ are positive definite matrices, and $(a_{ii})$ are the diagonal entries of $A$ , then

$\det A\circ B\geq\det B\,\prod{a_{ii}}$

with equality if and only if $A$ is a diagonal matrix.

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 $\sum_{i=1}^{\infty}a_{i}b_{i}$ .

References

1 R. A. Horn, C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, 1994.
2 V.V. Prasolov, Problems and Theorems in Linear Algebra, American Mathematical Society, 1994.
3 B. Mond, J. E. Pecaric, Inequalities for the Hadamard product of matrices, SIAM Journal on Matrix Analysis and Applications, Vol. 19, Nr. 1, pp. 66-70. http://epubs.siam.org/sam-bin/dbq/article/30295(link)

Title	Hadamard product
Canonical name	HadamardProduct
Date of creation	2013-03-22 14:15:28
Last modified on	2013-03-22 14:15:28
Owner	bbukh (348)
Last modified by	bbukh (348)
Numerical id	8
Author	bbukh (348)
Entry type	Definition
Classification	msc 15A15
Defines	Oppenheim inequality