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

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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
$$detA\circ B\ge detB\prod {a}_{ii}$$ 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 ${\sum}_{i=1}^{\mathrm{\infty}}{a}_{i}{b}_{i}$.
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. 6670. http://epubs.siam.org/sambin/dbq/article/30295(link)
Title  Hadamard product 

Canonical name  HadamardProduct 
Date of creation  20130322 14:15:28 
Last modified on  20130322 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 