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A Remark on a Result of Marvin Marcus

Published online by Cambridge University Press:  20 November 2018

N.S. Mendelsohn
Affiliation:
University of Manitoba
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Marcus [2] has proved the following theorem.

Suppose A is a non-negative normal matrix satisfying p(A) = 0 in which p(λ) is a monic polynomial no two of whose non-zero roots have the same modulus. Then there exists a permutation matrix P such that PAP* is a direct sum, PAP* = A1 ⊕ A2 ⊕ … ⊕ Am, in which each Ai is either O or primitive.

This note gives a generalisation of this result, dropping the non-negative assumption and weakening the normality assumption.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1963

References

1. Dulmage, A. L. and Mendelsohn, N. S. "The Characteristic Equation of an Imprimitive Matrix". Submitted to the Journal of S. I. A. M.Google Scholar
2. Marcus, Marvin "Another Remark on a Result of K. Goldberg". Can. Math. Bull. 6 (1963). p. 7.10.4153/CMB-1963-002-6Google Scholar