 
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 Hermitiancongruent 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 Hermitiancongruent. Moreover, when A and B are Hermitiancongruent,
we describe the possible inertias of the Hermitian matrices C that carry the congruence. We also give
necessary and sufficient conditions for any 2by2 nonsingular Hermitian matrices to be Hermitian
congruent. In both of the studied cases, we show that if A and B are real and Hermitiancongruent,
then they are congruent by a real symmetric matrix. Finally we note that if A and B are 2by2
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
