ON NONNEGATIVE MATRICES WITH A FULLY CYCLIC PERIPHERAL SPECTRUM

Authors

  • Bit-Shun Tam Department of Mathematics, Tamkang University, Tamsui, Taiwan 25137, Republic of China.

DOI:

https://doi.org/10.5556/j.tkjm.21.1990.4696

Keywords:

fully cyclic peripheral spectrum, reduced graph, M-matrix, basic class, nonnegative matrix, irreducible matrix, Perron-Frobenius eigenvalue, spectral radius

Abstract

Let $A$ be a square complex matrix. We denote by $\rho(A)$ the spectral radius of $A$. The set of eigenvalues of $A$ with modulus $\rho(A)$ is called the peripheral spectrum of $A$. The latter set is said to to be fully cyclic if whenever $\rho(A)\alpha x =Ax$, $x\neq 0$, $|a|= 1$, then $|x|(sgn \ x)^k$ is an eigenvector of $A$ corresponding to $\rho(A)\alpha^k$ for all integers $k$. In this paper we give some necessary conditions and a set of sufficient conditions for a nonnegative matrix to have a fully cyclic peripheral spectrum. Our conditions are given in terms of the reduced graph of a nonnegative matrix.

References

A Berman and R. J. Plemmons, Nonnegative Matrice, in the Mathematical Science,, Academic Press, New York, 1979.

C. D. H. Cooper, On the maximum eigenvalue of a reducible nonnegative real matrix, Math. Z. 13: 213-217 (1973).

R. A. Horn and C. R. Jolmson, Matrix Analysis, Cambridge Univ. Press, New York, 1985.

H. H. Schaefer, Banach Lattices and Positive Operator,, Springer-Verlag, New York 1974.

H. Schneider, The influence of the marked reduced graph of a nonnegative matrix on the Jordan form and on related properties: a survey, Linear Algebra Appl. 84: 161-189 {1986).

B. S. Tam and S. F. Wu, On the Collatz-Wielandt sets associated with a cone-preserving map, to appear in Linear Algebra Appl.

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Published

1990-03-01

How to Cite

Tam, B.-S. (1990). ON NONNEGATIVE MATRICES WITH A FULLY CYCLIC PERIPHERAL SPECTRUM. Tamkang Journal of Mathematics, 21(1), 65–70. https://doi.org/10.5556/j.tkjm.21.1990.4696

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Papers