A Note on Unique Solvability of the Generalized Absolute Value Matrix Equation

Abstract

In this survey paper, we focus on the necessary and sufficient conditions for the unique solvability and unsolvability of the absolute value equations (AVEs) during the last twenty years (2004 to 2023). We discussed unique solvability conditions for various types of AVEs like standard absolute value equation (AVE), Generalized AVE (GAVE), New generalized AVE (NGAVE), Triple AVE (TAVE) and a class of NGAVE based on interval matrix, P-matrix, singular value conditions, spectral radius and W-property. Based on unique solution of AVEs, we also discussed unique solvability conditions for linear complementarity problems (LCP) and horizontal linear complementarity problems (HLCP).

New sufficient conditions for the solvability of a new class of Sylvester-like absolute value matrix equation

“…The new Sylvester-like AVME (1) is equivalently written as NGAVE form (6). So now we use the results of the NGAVE for the new Sylvester-like AVME.…”

“…The proof of the above Theorem is directly held by Theorem 5 and Eq. (6). (iv) ((D −1 B) T ⊗ (AC −1 ) + I) is invertible and…”

“…In 2020, Dehghan et al [2] first considered the generalized absolute value matrix equations ( 4) and provided a matrix multi-splitting Picard-iterative method for solving the GAVME. In 2022, Kumar et al [5,6] provided two new conditions to ensure the unique solvability of the GAVME, the condition of Kumar et al [6] is superior to the conditions of Xie [20] and Dehghan et al [2]. In 2022, Tang et al [17] further discussed the unique solvability of the GAVME and provided a Picard-type method for the solution of the GAVME.…”