A note on the unique solvability condition for generalized absolute value matrix equation

Abstract

The spectral radius condition \[\rho (\vert A^{-1} \vert\cdot \vert B \vert)<1\]for the unique solvability of generalized absolute value matrix equation (GAVME) \[AX + B \vert X \vert = D\] is provided. For some instances, our condition is superior to the earlier published singular values conditions \(\sigma_{\max}(\vert B \vert)<\sigma_{\min}(A)\) [M. Dehghan, 2020] and \(\sigma_{\max}(B)<\sigma_{\min}(A)\) [Kai Xie, 2021]. For the validity of our condition, we also provided an example.

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

“…Above results of Theorem 5.5, Theorem 4.6, Theorem 5.7 and Theorem 5.8 are also extended for the AVME (see Theorem 1, Theorem 2, Theorem 3, Theorem 4 respectively [37] and [10,12]). Furthermore, Hashemi [6] and Wang et al [38] motivated from the results of Theorem 4.6, Theorem 4.11, Theorem 5.1 and Theorem 5.8 of GAVE, provided the unique solvability conditions for the Sylvesterlike absolute value matrix equation AXB + C|X|D 1 = E where A, B, C, D 1 , E, X are rectangular or square matrices of real entries.…”

“…In this article, based on the different matrix classes, we provide some new unique solvability conditions for GAVE. The unique solvability conditions of GAVE (1) are also helpful for obtaining the unique solvability conditions for the LCP and the GAVME [3,8,9,10]. Before designing an algorithm to solve a GAVE system, it is crucial to ensure the existence of a unique solution.…”

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

“…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.…”

“…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. In 2021, Hashemi [3] first considered the Sylvester-like absolute value matrix equations (5) and discussed its unique solvability conditions. Wang et al [18] provided new unique solvability conditions for the Sylvester-like AVME (5), which are different work from the Hashemi [3].…”

“…In 2021, Hashemi [3] first considered the Sylvester-like absolute value matrix equations (5) and discussed its unique solvability conditions. Wang et al [18] provided new unique solvability conditions for the Sylvester-like AVME (5), which are different work from the Hashemi [3]. Inspired by the above works on different types of matrix equations, Li [8] first considered the new class of Sylvester-like AVME (1) and provided unique solvability conditions for (1).…”