Generalized Perron roots and solvability of the absolute value equation
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[1] Technical University of Berlin
Technical University of Berlin
Berlin, Stadt, Alemania
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- Localización: Discrete Mathematics Days 2022 / Luis Felipe Tabera Alonso (ed. lit.), 2022, ISBN 978-84-19024-02-2, págs. 237-242
- Idioma: inglés
- Texto completo no disponible (Saber más ...)
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Resumen
Let A be a real (n × n)-matrix. The piecewise linear equation system z − A|z| = b iscalled an absolute value equation (AVE). It is well-known to be equivalent to the linearcomplementarity problem (LCP). For AVE and LCP unique solvability is comprehensivelycharacterized in terms of conditions on the spectrum (AVE), resp., the principal minors(LCP) of the coefficient matrix. For mere solvability no such characterization exists. Weclose this gap in the theory on the AVE-side. The aligning spectrum of A consists ofreal eigenvalues of the matrices SA, where S ∈ diag({±}n), which have a correspondingeigenvector in the positive orthant of Rn. For the mapping degree of the piecewise linearfunction z 7→ z − A|z| we prove, under some mild genericity assumptions on A: The degreeis 1 if all aligning values are smaller than 1, it is 0 if all aligning values are larger than 1,and in general it is congruent to (k + 1) mod 2 if k aligning values are larger than 1. Themodulus cannot be omitted because the degree can both increase and decrease.
