Matrix functions preserving sets of generalized nonnegative matrices

Abstract

Matrix functions preserving several sets of generalized nonnegative matrices are characterized. These sets include PFn, the set of n×n real eventually positive matrices; and WPFn, the set of matrices A ∈ Rn×n such that A and its transpose have the Perron-Frobenius property. Necessary conditions and sufficient conditions for a matrix function to preserve the set of n × n real eventually nonnegative matrices and the set of n × n real exponentially nonnegative matrices are also presented. In particular, it is shown that if f(0) 6= 0 and f′(0) 6= 0 for some entire function f, then such an entire function does not preserve the set of n×n real eventually nonnegative matrices. It is also shown that the only complex polynomials that preserve the set of n × n real exponentially nonnegative matrices are p(z) = az + b, where a,b ∈ R and a ≥ 0.