Generalized Perron Roots and Solvability of the Absolute Value Equation

Abstract

Let $A$ be a $n\times n$ real matrix. The piecewise linear equation system $z-A|z|=b$ 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 $F_A:{\mathbb{R}}^n\to {\mathbb{R}}^n,z\mapsto z-A|z|$ is congruent to $(k+1)\mathrm{mod}2$, 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.

20 pages, 2 figures

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