Hadamard product

DefinitionSuppose $A=(a_{ij})$ and $B=(b_{ij})$ are two $n\times m$-matriceswith entries in some field. Then their Hadamard product isthe 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 theHadamard product of ${\sum }_{i=1}^{\mathrm{\infty }}{a}_i$ and ${\sum }_{i=1}^{\mathrm{\infty }}{b}_i$ is${\sum }_{i=1}^{\mathrm{\infty }}{a}_i{b}_i$.