Spectral properties of sign symmetric matrices

Abstract

Spectral properties of sign symmetric matrices are studied.A criterion for sign symmetry of shifted basic circulant permutation matrices is proven, and is then used to answer the question which complex numbers can serve as eigenvalues of sign symmetric 3 × 3m atrices. The results are applied in the discussion of the eigenvalues of QM-matrices.In particular, it is shown that for every positive integer n there exists a QM-matrix A such that A k is a sign symmetric P -matrix for all k ≤ n, but not all the eigenvalues of A are positive real numbers. The research of the relationship between stability, positivity of principal minors and sign symmetry was motivated by a research problem by Taussky (12) calling for investigation of the common properties of totally positive matrices, nonsingular M - matrices and positive definite matrices. Stability, positivity of principal minors and weak sign symmetry are amongst those common properties. This paper deals with spectral properties of general sign symmetric matrices and of sign symmetric matrices having some additional properties.