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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

Powers of matrices with positive definite real part


Author: Charles R. Johnson
Journal: Proc. Amer. Math. Soc. 50 (1975), 85-91
MSC: Primary 15A48
MathSciNet review: 0369395
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Abstract: For $ n$ by $ n$ complex matrices $ A$ the following two facts are proven by elementary techniques: 1. If $ {A^m}$ is never normal, $ m \in {I^ + }$, then the equation $ x{A^m}{x^\ast } = 0$ has a solution $ 0 \ne x \in {C^n},m \in {I^ + }$; 2. If $ H(A) = (A + {A^\ast })/2$ is positive definite, then $ H({A^m})$ is positive definite for all $ m \in {I^ + }$ if and only if $ A$ is Hermitian.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1975-0369395-7
PII: S 0002-9939(1975)0369395-7
Keywords: Hermitian matrix, positive definite, normal matrix, field of values
Article copyright: © Copyright 1975 American Mathematical Society