Congruence of Hermitian matrices by Hermitian matrices

• Type: Article in International Scientific Journal
• Date: 2007

Authors

Maria Isabel Bueno (Author) Other; Susana Borges Furtado (Author) FEP; Charles Johnson (Author) Other

Journal

Title: Linear Algebra and Its Applications

Vol. 425

Pages: 63-76

ISSN: 0024-3795

Publisher: Elsevier

Indexing

ISI Web of Knowledge - 4 Citations

ISI Web of Science

Scopus - 6 Citations

Scientific classification

FOS: Natural sciences

Other information

Authenticus ID: P-004-8K7

DOI: 10.1016/j.laa.2007.03.016

Abstract (EN): Two Hermitian matrices A, B E Mu (C) are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix C is an element of M-n (C) such that B = CAC. In this paper, we give necessary and sufficient conditions for two nonsingular simultaneously unitarily diagonalizable Hermitian matrices A and B to be Hermitian-congruent. Moreover, when A and B are Hermitian-congruent, we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give necessary and sufficient conditions for any 2-by-2 nonsingular Hermitian matrices to be Hermitian-congruent. In both of the studied cases, we show that if A and B are real and Hermitian-congruent, then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2-by-2 nonsingular real symmetric matrices having the same sign pattern, then there is always a real symmetric matrix C satisfying B = CAC. Moreover, if both matrices are positive, then C can be picked with arbitrary inertia.

Language: English

Type (Professor's evaluation): Scientific

No. of pages: 14