Generalized Perron Roots and Solvability of the Absolute Value Equation

TL;DR

This paper introduces the aligned spectrum of a matrix to characterize the solvability of the absolute value equation (AVE), extending existing theory beyond unique solutions and providing formulas for the mapping degree related to AVE and LCP.

Contribution

It defines the aligned spectrum of a matrix and proves its relation to the mapping degree of the AVE function, broadening the understanding of solvability conditions.

Findings

Mapping degree of AVE function relates to the aligned spectrum.

Derived formulas connect the aligned spectrum with solvability.

Extended characterization to non-unique solutions of AVE.

Abstract

Let $A$ be a $n\times n$ real matrix. The piecewise linear equation system $z-A\vert z\vert =b$ 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 $A$. For mere, possibly non-unique, solvability no such characterization exists. We narrow this gap in the theory. That is, we define the concept of the aligned spectrum of $A$ and prove, under some mild genericity assumptions on $A$, that the mapping degree of the piecewise linear function $F_A:\mathbb{R}^n\to\mathbb{R}^n\,, z\mapsto z-A\lvert z\rvert$ is congruent to $(k+1)\mod 2$, where $k$ is the number of aligned values of $A$ which are larger than $1$. We also derive an exact--but more technical--formula for the degree of $F_A$ in terms of the aligned spectrum. Finally, we derive the analogous quantities and results for the LCP.