Abstract
The generalized absolute value matrix equation has application in a variety of optimization problems, its unique solvability is still on the way. In this note, the unique solvability of the generalized absolute value matrix equation is considered. A new unique solvability of generalized absolute value matrix equation is given. The obtained result can be regarded as an extension of the absolute value equation to the generalized absolute value matrix equation. As an application, new convergence of matrix multisplitting Picard-iterative method is presented.
Highlights
In this note, the following generalized absolute value matrix equation (GAVME) is considered + | |=, (1)where, ∈ × are given matrices and ∈ × is an unknown matrix, | | denotes the component-wise absolute value of the matrix, i.e., | | =
It is understood that only Dehghan and Shirilord [23] have provided the condition of unique solvability for the GAVME (1) and proposed the matrix multisplitting Picard-iterative method for solving (1) recently
Kai Xie: On the Unique Solvability of the Generalized Absolute Value Matrix Equation be invalid to judge the unique solution of GAVME in some case, and give a new unique solvability for the GAVME (1), which is weaker than that in [23], and is same as that for generalized absolute value equation (GAVE) (2) in [11]
Summary
The following generalized absolute value matrix equation (GAVME) is considered. The GAVME (1) is a generalization of the absolute value equation (AVE). It is understood that only Dehghan and Shirilord [23] have provided the condition of unique solvability for the GAVME (1) and proposed the matrix multisplitting Picard-iterative method for solving (1) recently. Kai Xie: On the Unique Solvability of the Generalized Absolute Value Matrix Equation be invalid to judge the unique solution of GAVME in some case, and give a new unique solvability for the GAVME (1), which is weaker than that in [23], and is same as that for GAVE (2) in [11]. As an application of new result, the convergence of matrix multisplitting Picard-iterative method for solving (1) is restated
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