Some determinantal inequalities for Hadamard product of matrices

Abstract

The main result of this paper is the following: if both A=(a ij) and B=(b ij) are M-matrices or positive definite real symmetric matrices of order n, the Hadamard product of A and B is denoted by A°B, and A k and B k (k=1,2,⋯,n) are the k×k leading principal submatrices of A and B, respectively, then det(A°B) ≥ det(AB)∏ (k=2)(n) (a kk det A k-1/detA k)+ b kkdetB k-1/detB k-1). © 2003 Elsevier Science Inc. All rights reserved.