A primer of Perron–Frobenius theory for matrix polynomials

Abstract

We present an extension of Perron–Frobenius theory to the spectra and numerical ranges of Perron polynomials, namely, matrix polynomials of the form L(λ)=Iλ m−A m−1λ m−1−⋯−A 1λ−A 0, where the coefficient matrices are entrywise nonnegative. Our approach relies on the companion matrix linearization. First, we recount the generalization of the Perron–Frobenius Theorem to Perron polynomials and report some of its consequences. Subsequently, we examine the role of L( λ) in multistep difference equations and provide a multistep version of the Fundamental Theorem of Demography. Finally, we extend Issos' results on the numerical range of nonnegative matrices to Perron polynomials.