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 $zA\vert z\vert =b$ is called an absolute value equation (AVE). It is wellknown 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 signreal spectral radius of $A$. 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 $A$ and prove, under some mild genericity assumptions on $A$, that the mapping degree of the piecewise linear function $F_A:\mathbb{R}^n\to\mathbb{R}^n\,, z\mapsto zA\lvert z\rvert$ 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 exactbut more technicalformula for the degree of $F_A$ in terms of the aligned spectrum. Finally, we derive the analogous quantities and results for the LCP.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 DOI:
 10.48550/arXiv.1912.08157
 arXiv:
 arXiv:1912.08157
 Bibcode:
 2019arXiv191208157R
 Keywords:

 Mathematics  Optimization and Control;
 Mathematics  Numerical Analysis;
 65K99;
 90C33;
 15A24
 EPrint:
 20 pages, 2 figures