Perron-Frobenius theory for complex matrices

Abstract

The purpose of this paper is to present a unified Perron-Frobenius Theory for nonnegative, for real not necessarily nonnegative and for general complex matrices. The sign-real spectral radius was introduced for general real matrices. This quantity was shown to share certain properties with the Perron root of nonnegative matrices. In this paper we introduce the sign-complex spectral radius. Again, this quantity extends many properties of the Perron root of nonnegative matrices to general complex matrices. Various characterizations will be given, and many open problems remain.