Home

About

Advanced Search

Browse by Discipline

Scientific Societies

E-print Alerts

Add E-prints

E-print Network
FAQHELPSITE MAPCONTACT US


  Advanced Search  

 
CONGRUENCE OF HERMITIAN MATRICES BY HERMITIAN M.I. BUENO, S. FURTADO , AND C.R. JOHNSON
 

Summary: CONGRUENCE OF HERMITIAN MATRICES BY HERMITIAN
MATRICES
M.I. BUENO, S. FURTADO , AND C.R. JOHNSON§
Abstract. Two Hermitian matrices A, B Mn(C) are said to be Hermitian-congruent if there
exists a nonsingular Hermitian matrix C Mn(C) such that B = CAC. In this paper, we give neces-
sary 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 sym-
metric matrix C satisfying B = CAC. Moreover, if both matrices are positive, then C can be picked
with arbitrary inertia.
Key words. Congruence, Hermitian matrix, simultaneously unitarily diagonalizable, sign pat-
tern.
AMS subject classifications. 15A21, 15A24, 15A48, 15A57
1. Introduction. Matrices A, B Mn(C) (Mn for short) are said to be congru-
ent if there is a nonsingular matrix C Mn(C) such that B = C
AC. Congruence

  

Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara

 

Collections: Mathematics