Generalized Perron Roots and Solvability of the Absolute Value Equation
Manuel Radons, Josué Tonelli-Cueto · SIAM Journal on Matrix Analysis and Applications · 2023
Abstract. Let [Formula: see text] be an [Formula: see text] real matrix. The piecewise linear equation system [Formula: see text] is called an absolute value equation (AVE). It is well known to be equivalent to the linear complementarity problem. Unique solvability of the AVE is known to be characterized in terms of a generalized Perron root called the sign-real spectral radius of [Formula: see text]. For mere, possibly nonunique, solvability no such characterization exists. We narrow this gap in the theory. That is, we define the concept of the aligned spectrum of [Formula: see text] and prove, under some mild genericity assumptions on [Formula: see text], that the mapping degree of the piecewise linear function [Formula: see text] is congruent to [Formula: see text], where [Formula: see text] is the number of aligned values of [Formula: see text] which are larger than 1. We also derive an exact—but more technical—formula for the degree of [Formula: see text] in terms of the aligned spectrum. Finally, we derive the analogous quantities and results for the linear complementarity problem.