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Hadamard product
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(Definition)
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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 $a_{ij} b_{ij}$ .
Suppose $A,B,C$ are matrices of the same size and $\lambda$ is a scalar. Then \begin{eqnarray*} A\circ B &=& B\circ A, \\ A\circ (B+C) &=& A\circ B + A\circ C, \\ A\circ (\lambda B) &=& \lambda (A\circ B), \end{eqnarray*}
- 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 $$\det A\circ B \ge \det B\,\prod{a_{ii}}$$ with equality if and only if $A$ is a diagonal matrix.
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 $\sum_{i=1}^\infty a_i b_i$ .
- 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)
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"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 21158 times total.
Classification:
| AMS MSC: | 15A15 (Linear and multilinear algebra; matrix theory :: Determinants, permanents, other special matrix functions) |
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Pending Errata and Addenda
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