PlanetMath: Hadamard product

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Hadamard product

(Definition)

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 and , that is, the $n\times m$ -matrix $A\circ B$ whose th entry is $a_{ij} b_{ij}$ .

Properties

Suppose

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 are diagonal matrices, then $A\circ B=AB$ .
(Oppenheim inequality) [2]: If are positive definite matrices, and $(a_{ii})$ are the diagonal entries of , then
$\displaystyle \det A\circ B \ge \det B\,\prod{a_{ii}}$
with equality if and only if 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 $\sum_{i=1}^\infty a_i b_i$ .

Bibliography

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. (link)

"Hadamard product" is owned by bbukh. [ owner history (1) ]

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Also defines:

Oppenheim inequality

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Cross-references: power series, equality, diagonal, positive definite, diagonal matrices, scalar, size, matrices, product, field
There are 4 references to this entry.

This is version 5 of Hadamard product, born on 2004-03-14, modified 2005-10-28.
Object id is 5706, canonical name is HadamardProduct.
Accessed 15453 times total.

Classification:

AMS MSC:

15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions)

Pending Errata and Addenda

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