Mathematics Faculty Articles

Title

Schur Complements and Matrix Inequalities of Hadamard Products

Document Type

Article

Publication Date

1997

Publication Title

Linear and Multilinear Algebra

Keywords

Schur complement, Hadamard product, Principal submatrix

ISSN

0308-1087

Volume

43

Issue/No.

1-3

First Page

315

Last Page

326

Abstract

Let A and B be n-square positive definite matrices. Denote the Hadamard product of A and B by A o B. The main results of the paper are:

1. For any matrices C and D of size m×n

(C ο D)(A ο B)-1 (C ο D)* ≤ (CA-1 C*) ο (DB-1D*)

and

2. Let A/α be the Schur complement of A(α) in A. Then

(A ο B)/α ≥ A/α ο B/α.

Some other matrix inequalities of Schur complements and Hadamard products of positive definite matrices are also presented.

DOI

10.1080/03081089708818531

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Peer Reviewed

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