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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
DOI: https://doi.org/10.1090/S0002-9939-1975-0369395-7
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.


References [Enhancements On Off] (What's this?)

  • [1] W. F. Donoghue, On the numerical range of a bounded operator, Michigan Math. J. 4 (1957), 261-263. MR 20 #2622. MR 0096127 (20:2622)
  • [2] C. R. Johnson and M. Newman, Triangles generated by powers of triplets on the unit circle, J. Res. Nat. Bur. Standards Sect. B 77B (1973), 137-142. MR 0389831 (52:10661)
  • [3] H. Weyl, Über die Gleichverteilung von Zahlen mod Eins, Math. Ann. 77 (1916), 313-352. MR 1511862

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0369395-7
Keywords: Hermitian matrix, positive definite, normal matrix, field of values
Article copyright: © Copyright 1975 American Mathematical Society

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