Spectral properties of sign symmetric matrices

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 × 3 matrices. 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.