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 $A\circ B=AB$.

• (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 HadamardProduct 2013-03-22 14:15:28 2013-03-22 14:15:28 bbukh (348) bbukh (348) 8 bbukh (348) Definition msc 15A15 Oppenheim inequality