Generalisation of the Perron–Frobenius theory to matrix pencils

Abstract

We present a new extension of the well-known Perron-Frobenius theorem to regular matrix pairs (E, A). The new extension is based on projector chains and is motivated from the solution of positive differential-algebraic systems or descriptor systems. We present several examples where the new condition holds, whereas conditions in previous literature are not satisfied.

Singularity in the Discrete Dynamic Leontief Model

“…is the Drazin inverse of B , matrix R is an invertible matrix such that matrix C is invertible, and matrix N is nilpotent with m = index N. Using these notations system (2) can be divided into a forward and a backward recursive system: Now we turn to the solution of (1') and its difference equation form (2) by using the Weierstrass canonical form (Gantmacher, 1959;Mehrmann et al, 2008). In this approach we use the Campbell regularity condition and the solution takes a form similar to the solution in Campbell's paper.…”

Stability of switched positive descriptor systems with average dwell time switching

“…The matrix products do not depend on the specific choice ofλ (see Ref. [8] for more details). Hence, to calculate conveniently, we can chooseλ = 0, (2) and (3) depend on υ 0 ,Ê i andÊ D i .…”

“…Under this assumption, the stability issue has been addressed in Ref. [6] by using a generalized Perron-Frobenius type condition [8] , and in particular, a Lyapunov-type stability condition is derived. In Refs.…”

The level set method for the two-sided max-plus eigenproblem

“…Problem (1) is related to the Perron-Frobenius theory for the two-sided eigenproblem in the conventional linear algebra, as studied in [34,35]. When both matrices are nonnegative and depend on a large parameter, it can be shown following the lines of [1,Theorem 1] that the asymptotics of an eigenvalue with nonnegative eigenvector is controlled by an eigenvalue of (1).…”