Open AccessON NONNEGATIVE MATRICES WITH A FULLY CYCLIC PERIPHERAL SPECTRUM
Author(s) -
Bit-Shun Tam
Publication year - 1990
Publication title -
tamkang journal of mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.324
H-Index - 18
eISSN - 0049-2930
pISSN - 2073-9826
DOI - 10.5556/j.tkjm.21.1990.4696
Subject(s) - mathematics , eigenvalues and eigenvectors , spectral radius , combinatorics , spectrum (functional analysis) , matrix (chemical analysis) , graph , modulus , physics , geometry , quantum mechanics , materials science , composite material
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.