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Generalized Perron Roots and Solvability of the Absolute Value Equation

Manuel Radons, Josué Tonelli-Cueto

TL;DR

The paper develops a topological degree framework for the absolute value equation $z-A|z|=b$ by introducing the aligned spectrum $ ext{Spec}^{a}(A)$ in addition to the classical sign-real spectrum. It proves that, for generic $A$ with $1 otin ext{Spec}^{a}(A)$, the degree $ ext{deg}\,F_A$ is determined modulo 2 by the number of aligned values greater than 1, and it provides an exact formula in terms of signed contributions from aligned triples. These AVE results extend to LCPs via the map $G_M$, yielding parity-based solvability criteria and a direct translation of aligned-spectrum ideas. The work also shows that generic perturbations preserve degree while enabling clear counting via fixed points on the sphere, enabling practical sufficient conditions for solvability and insights into the structure of $Q$-matrices. Overall, the paper links generalized Perron-type spectral data to solvability through degree theory for AVEs and LCPs, with an emphasis on robustness under perturbations and exact accounting of alignment structure.

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.

Generalized Perron Roots and Solvability of the Absolute Value Equation

TL;DR

The paper develops a topological degree framework for the absolute value equation by introducing the aligned spectrum in addition to the classical sign-real spectrum. It proves that, for generic with , the degree is determined modulo 2 by the number of aligned values greater than 1, and it provides an exact formula in terms of signed contributions from aligned triples. These AVE results extend to LCPs via the map , yielding parity-based solvability criteria and a direct translation of aligned-spectrum ideas. The work also shows that generic perturbations preserve degree while enabling clear counting via fixed points on the sphere, enabling practical sufficient conditions for solvability and insights into the structure of -matrices. Overall, the paper links generalized Perron-type spectral data to solvability through degree theory for AVEs and LCPs, with an emphasis on robustness under perturbations and exact accounting of alignment structure.

Abstract

Let be a real matrix. The piecewise linear equation system 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 . 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 and prove, under some mild genericity assumptions on , that the mapping degree of the piecewise linear function is congruent to , where is the number of aligned values of which are larger than . We also derive an exact--but more technical--formula for the degree of in terms of the aligned spectrum. Finally, we derive the analogous quantities and results for the LCP.

Paper Structure

This paper contains 14 sections, 28 theorems, 107 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Sign-Real and Aligned Spectra
  3. Sign-real spectrum and bijectivity
  4. Aligned spectrum and the eigenproblem
  5. Degree of a map
  6. Generic matrices
  7. Interpretation of count for generic matrices
  8. Perturbation of matrices to make them generic
  9. Transfer of results to LCPs
  10. Proof of Theorems \ref{['thm:degswitch']} and \ref{['thm:exactformula']} and Corollaries \ref{['cor:numberalignedvalues']} and \ref{['cor:solvabilityAVE']}
  11. Proof of Theorem \ref{['thm:degswitch']}
  12. Proof of Corollary \ref{['cor:numberalignedvalues']}
  13. Proof of Corollary \ref{['cor:solvabilityAVE']}
  14. Proof of Theorem \ref{['thm:exactformula']}

Key Result

Theorem 1.1

Let $A\in\operatorname{M}_n(\mathbb R)$ be generic such that $1\notin \mathop{\mathrm{\operatorname{Spec}^a}}\nolimits(A)$. Then the degree of $F_A$ is well-defined and it satisfies that Moreover, $\deg F_A$ equals $1$ if all aligned values are smaller than $1$, and it equals $0$ if all aligned values are larger than $1$.

Figures (2)

  • Figure 1: Image of the unit circle under $F_{B_\varepsilon}$. For $\varepsilon=-0.01$ (left) and $\varepsilon=0.01$ (right) it winds around the origin.
  • Figure 2: Image of the unit circle under $F_{A_\varepsilon}$. For $\varepsilon=0.2$ it winds around the origin (left). For $\varepsilon=-0.2$ it does not (right).

Theorems & Definitions (59)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Definition 2.1
  • Theorem 2.2
  • Proposition 2.3
  • proof
  • Definition 2.4
  • ...and 49 more