The singular values of a Hadamard product: a basic inequality
- 1 December 1987
- journal article
- research article
- Published by Taylor & Francis in Linear and Multilinear Algebra
- Vol. 21 (4) , 345-365
- https://doi.org/10.1080/03081088708817810
Abstract
For any r- by n complex matrixZ, let c 1(Z) ⩾ … ⩾ cn (Z) ⩾ 0 denote the Euclidean lengths of the columns of Z, arranged in descending order. Denote the similarly ordered singular vaiucs of any n-by-n complex matrix C by σ1 (C) ⩾ … ⩾ σ n (C) ⩾ 0. Let A and B be given n-by-n complex matrices and write the Hadamard (entry-wise) product of A and B as A B We show that for any r-by-n complex matrices X and Y such that A = X * Y. Several recently proved inequalities and some classical inequalities for Hadamard products follow immediately from this result.Keywords
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