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Congruence of Hermitian matrices by Hermitian matrices
- Source :
-
Linear Algebra & its Applications . Aug2007, Vol. 425 Issue 1, p63-76. 14p. - Publication Year :
- 2007
-
Abstract
- Abstract: Two Hermitian matrices are said to be Hermitian-congruent if there exists a nonsingular Hermitian matrix such that . 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 . Moreover, if both matrices are positive, then C can be picked with arbitrary inertia. [Copyright &y& Elsevier]
- Subjects :
- *MATRICES (Mathematics)
*UNIVERSAL algebra
*SYMMETRIC matrices
*LINEAR algebra
Subjects
Details
- Language :
- English
- ISSN :
- 00243795
- Volume :
- 425
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Linear Algebra & its Applications
- Publication Type :
- Academic Journal
- Accession number :
- 25345230
- Full Text :
- https://doi.org/10.1016/j.laa.2007.03.016