The Unique Solvability Conditions for the Generalized Absolute Value Equations

TL;DR

This paper establishes conditions for the unique solvability and unsolvability of generalized absolute value equations and matrix equations, providing theoretical insights into their solution properties.

Contribution

It introduces new solvability criteria for GAVE and GAVME, extending understanding of when these equations have unique solutions.

Findings

Derived necessary and sufficient conditions for unique solvability.

Extended solvability analysis to matrix equations.

Discussed aspects of solvability and unsolvability of AVE.

Abstract

This paper investigates the conditions that guarantee unique solvability and unsolvability for the generalized absolute value equations (GAVE) given by $Ax - B \vert x \vert = b$. Further, these conditions are also valid to determine the unique solution of the generalized absolute value matrix equations (GAVME) $AX - B \vert X \vert =F$. Finally, certain aspects related to the solvability and unsolvability of the absolute value equations (AVE) have been deliberated upon.