# Generalized Perron Roots and Solvability of the Absolute Value Equation

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 uniquely solvable for all b∈R^n if and only if a quantity called the sign-real spectral radius of A is smaller than one. We construct a similar quantity that we call the aligning spectral radius ρ^a of A. We prove that the AVE is solvable for all b∈R^n if the aligning spectral radius of A is smaller than one. For n≤ 2 this condition is also necessary. Structural properties of the aligning spectral radius are investigated.

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