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Article Dans Une Revue SIAM Journal on Matrix Analysis and Applications Année : 2023

Generalized Perron Roots and Solvability of the Absolute Value Equation

Résumé

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 close 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.
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hal-03738197 , version 1 (25-07-2022)

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Manuel Radons, Josué Tonelli-Cueto. Generalized Perron Roots and Solvability of the Absolute Value Equation. SIAM Journal on Matrix Analysis and Applications, 2023, 44 (4), pp.1645-1666. ⟨10.1137/22M1517184⟩. ⟨hal-03738197⟩
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