The Perron eigenspace of nonnegative almost skew-symmetric matrices and Levinger’s transformation

Abstract

Let A be a nonnegative square matrix whose symmetric part has rank one. Tournament matrices are of this type up to a positive shift by 1/2I. When the symmetric part of A is irreducible, the Perron value and the left and right Perron vectors of L(A,α)=(1−α)A+αAt are studied and compared as functions of α∈[0,1/2]. In particular, upper bounds are obtained for both the Perron value and its derivative as functions of the parameter α via the notion of the q-numerical range.