Powers of matrices with positive definite real part
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- by Charles R. Johnson PDF
- Proc. Amer. Math. Soc. 50 (1975), 85-91 Request permission
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
- William F. Donoghue Jr., On the numerical range of a bounded operator, Michigan Math. J. 4 (1957), 261–263. MR 96127
- Charles R. Johnson and Morris Newman, Triangles generated by powers of triplets on the unit circle, J. Res. Nat. Bur. Standards Sect. B 77B (1973), no. 3-4, 137–141. MR 389831
- Hermann Weyl, Über die Gleichverteilung von Zahlen mod. Eins, Math. Ann. 77 (1916), no. 3, 313–352 (German). MR 1511862, DOI 10.1007/BF01475864
Additional Information
- © Copyright 1975 American Mathematical Society
- 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