Generalized Perron roots and solvability of the absolute value equation
Abstract
Let A be a real (n × n)-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 (LCP). For AVE and LCP unique solvability is comprehensively characterized in terms of conditions on the spectrum (AVE), resp., the principal minors (LCP) of the coefficient matrix. For mere solvability no such characterization exists. We close this gap in the theory on the AVE-side. The aligning spectrum of A consists of real eigenvalues of the matrices SA, where S ∈ diag({±}n), which have a corresponding eigenvector in the positive orthant of Rn. For the mapping degree of the piecewise linear function z → z − A|z| we prove, under some mild genericity assumptions on A: The degree is 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. The modulus cannot be omitted because the degree can both increase and decrease.
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