You are here
HomeHadamard product
Primary tabs
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
$\displaystyle A\circ B$  $\displaystyle=$  $\displaystyle B\circ A,$  
$\displaystyle A\circ(B+C)$  $\displaystyle=$  $\displaystyle A\circ B+A\circ C,$  
$\displaystyle A\circ(\lambda B)$  $\displaystyle=$  $\displaystyle\lambda(A\circ B),$ 

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. 6670. (link)
Mathematics Subject Classification
15A15 no label found Forums
 Planetary Bugs
 HS/Secondary
 University/Tertiary
 Graduate/Advanced
 Industry/Practice
 Research Topics
 LaTeX help
 Math Comptetitions
 Math History
 Math Humor
 PlanetMath Comments
 PlanetMath System Updates and News
 PlanetMath help
 PlanetMath.ORG
 Strategic Communications Development
 The Math Pub
 Testing messages (ignore)
 Other useful stuff
 Corrections
Comments
standard symbol for Hadamard product (elebyele mult)?
Is there a standard symbol for the Hadamard product?
This entry uses the centered open dot, and I've seen other references using that too. But then I've also seen the circledot used in print, or the heavy solid centered dot, or I've even seen folks (including myself) invent their own symbol.
Thanks. Sorry such a mindless question.