Matrices with Sign Symmetric Diagonal Shifts or Scalar Shifts

Abstract

We generalize the concepts of sign symmetry and weak sign symmetry by defining k-sign symmetric matrices. For a positive integer k, we show that all diagonal shifts of an irreducible matrix are k-sign symmetric if and only if the matrix is diagonally similar to a Hermitian matrix. A similar result holds for scalar shifts, but requires an additional condition in the case $k = 1$. Extensions are given to reducible matrices.