The singular values of a Hadamard product: a basic inequality

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.