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 formwhere 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.

Perron–Frobenius theorems for the numerical range of semi-monic matrix polynomials

“…For a single matrix A 0 with nonnegative entries the Perron-Frobenius theory is applied to the spectrum (= set of eigenvalues; see [11,18]) and the numerical range (see [13,17,19]). One of the main results of this theory says that the peripheral part of the spectrum and the numerical range of an irreducible entrywise nonnegative matrix have the same cyclic properties, the proof thereof relying on Wielandt's Lemma.…”

“…Properties of the numerical range of monic polynomials with entrywise nonnegative coefficients were considered in [19]. The proofs are based on the linearisation by the companion matrix C Q and by applying the Perron-Frobenius theory to C Q .…”

Perron‐Frobenius type results for the block numerical range

“…Previous results in this direction can be found in [2] and [3] where the numerical range of positive matrices was studied. For an investigation of the block numerical range of positive matrices we refer to [1], and for the numerical range of matrix polynomials we refer to [1] and [5].…”

Generalized Perron–Frobenius Theorem for Multiple Choice Matrices, and Applications

“…We note that whereas all generalizations concern the Perron-Frobenius property, the Collatz-Wielandt property is not always established. The long series of existing PF extensions includes [22,13,30,18,32,19,28,21]. We next discuss these extensions in comparison to the current work.…”