Schur Complements and Matrix Inequalities of Hadamard Products

Authors

Bo-Ying Wang, Beijing Normal University
Fuzhen Zhang, Nova Southeastern UniversityFollow

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.

Comments

AMS Subject: 15A45

NSUWorks Citation

Wang, Bo-Ying and Zhang, Fuzhen, "Schur Complements and Matrix Inequalities of Hadamard Products" (1997). Mathematics Faculty Articles. 106.
https://nsuworks.nova.edu/math_facarticles/106

DOI

10.1080/03081089708818531