On the Perron–Frobenius theory for complex matrices

Abstract

We extend here the Perron–Frobenius theory of nonnegative matrices to certain complex matrices. Following the generalization of the Perron–Frobenius theory to matrices that have some negative entries, given by Noutsos [14], we introduce here two types of extensions of the Perron–Frobenius theory to complex matrices. We present and prove here some sufficient conditions and some necessary and sufficient conditions for a complex matrix to have a Perron–Frobenius eigenpair. We apply this theory by introducing Perron–Frobenius splittings, as well as complex Perron–Frobenius splittings, for the solution of complex linear systems Ax=b, by classical iterative methods. Perron–Frobenius splittings constitute an extension of the well-known regular splittings, weak regular splittings and nonnegative splittings. Convergence and comparison properties are also given and proved.