Hadamard product

Definition Suppose A=(aij) and B=(bij) are two n×m-matrices with entries in some field. Then their Hadamard productMathworldPlanetmath is the entry-wise product of A and B, that is, the n×m-matrix AB whose (i,j)th entry is aijbij.


Suppose A,B,C are matrices of the same size and λ is a scalar. Then

AB = BA,
A(B+C) = AB+AC,
A(λB) = λ(AB),
  • If A,B are diagonal matricesMathworldPlanetmath, then AB=AB.

  • (Oppenheim inequality) [2]: If A,B are positive definite matrices, and (aii) are the diagonal entries of A, then


    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 i=1ai and i=1bi is i=1aibi.


  • 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
Canonical name HadamardProduct
Date of creation 2013-03-22 14:15:28
Last modified on 2013-03-22 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