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

$A\circ B$ $=$ $B\circ A,$

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

$A\circ (\lambda B)$ $=$ $\lambda (A\circ B),$

• If $A,B$ are diagonal matrices, then $A\circ B=AB$.

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

  $$detA\circ B\ge detB\prod a_{ii}$$

  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 ${\sum }_{i=1}^{\mathrm{\infty }}{a}_i$ and ${\sum }_{i=1}^{\mathrm{\infty }}{b}_i$ is ${\sum }_{i=1}^{\mathrm{\infty }}{a}_i{b}_i$.

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