Eigenvalue frequency and consistent sign pattern matrices
Carolyn A. Eschenbach, Frank J. Hall, Zhongshan Li · Czechoslovak Mathematical Journal · 1994
An n x n sign pattern matrix A is fc-consistent if every real matrix whose sign pattern is indicated by A has fc real eigenvalues and n -fc nonrcal eigenvalues.First we establish several general properties of fc-consistent patterns, for any integer fc, where 0 ^ fc ^ ?i.We then characterize patterns that are permutation similar to an irreducible, sign symmetric, tridiagonal matrix.Further, we establish a graph theoretic necessary condition for irreducible, tridiagonal patterns to be consistent, and we relate this condition to the cycle structure of the matrix.Finally we provide other interesting classes of consistent sign patterns.