Matrix Inverse Problemunder the Central Master Submatrix Constraint AX = B of the Generalized Centersymmetric Solution

This paper begins by identifying a problem related to the solvability of the matrix equation AX=B, particularly within the realm of generalized centrosymmetric matrices. It exploits the distinctive properties inherent in generalized centrosymmetric matrices, primary focus involves investigating their solutions within the context of the matrix equation AX=B. This paper establishes the necessary and sufficient conditions for solvability while formulating a comprehensive expression for the generalized centrosymmetric solution. Furthermore, the paper delves into addressing the associated optimal approximation quandary concerning a specified matrix within the solution set.


Introduction
The matrix inverse problem B AX  , here X , B stands as a pivotal subset of inverse problems concerning matrices constrained by submatrices.Over the recent years, a multitude of scholars has dedicated substantial focus to this domain, resulting in a series of notable achievements.Authors Zhao [1] obtained bisymmetric solutions of matrix equations B AX  under centural principle submatrix,Authors Guo and Zhou [2][3] studied a class of inverse problems with centrosymmetric and anticentro symmetric of matrix equations B AX  , respectively, and Authors Xu,Zhao ,Chen and Z [4-8] solved matrix equations with different constraints, respectively, and obtained very good results.This study delves deeply into the domain of generalized centrosymmetric solutions under the centural principle submatrix constraint for Matrix inverse problems B AX  and their optimal approximation problems.
The notation used throughout this manuscript is presented as follows: Within the scope of this manuscript, we investigate the following pair of challenges: Problem 1: Knowing that matrices Here A S denotes the solution set for Problem 1.

Solution of Problem 1
Initially, we provide the subsequent lemma. where In particular, when Then we have , have the expression (7), then the k order centered principal subarray of A has the potential to be formulated as .
have the form of (1) and Then a sufficient condition for 0 A to be the central principal subarray of By the expression for generalized centrosymmetric matrix and Lemma 2.3, even-order matrices have only even-order central principal subarrays, and we have , by the expression of the generalized centrosymmetric matrix, the odd order matrix has only odd order centrosymmetric principal subarrays, and we have From the above prove , it follows that the k order-centered principal subarray 0 A of the structure of the generalized centrosymmetric matrix A aligns with that of (3), with the sufficient condition as stated in the theorem.

Lemma 2.5[1]
Let of generalized form can be tabulated as and the sufficient condition for , and the generalization can still be expressed as (4).
Lemma 2.6 [1] Given and the sufficient condition for 2

S
to be nonempty for 0 and the generalization can still be expressed as (5).
Lemma 2.7 [6] Given where Then the sufficient condition for problem 1 to have a solution is that and the constituents within set A S , denoted by A , have the capacity to be articulated as. where

A and 22
A satisfy . By Lemma 2.4, the sufficient condition for equation (13) to hold is that i.e., equation ( 13) holds.In this case 11 A and 22 A can be expressed as where are arbitrary.Substituting Equation (19) into Equation ( 18) and noting Equation (12), we have By Lemma 2.5, the above equation holds if and only if )( , 0 ( , Thus the above equation is equivalent to ( 14), and at this point Thus the resolution to Problem 1 holds the capacity to be articulated as (15).

Existence of Solutions to Problem 2
When there is a solution to Problem 1, verifying A S as a closed convex set is straightforward, leading to the existence of a singular optimal approximate solution * A for Problem 2. Herein, we furnish explicit formulations for * A .

Theorem 3.1 Given
If there is a solution to Problem 1, consequently, there exists a singular optimal approximate solution, denoted as * A , to Problem 2, which has the potential to be formulated as where .

Conclusion
The focus of this article is on exploring generalized centrosymmetric solutions within a subset of matrix equations B AX  .Subsequently, we examine the fundamental requirements for the existence of solutions and present optimal approximation methods, adhering to the Frobenius paradigm for any given matrix.The methods we use focus on matrix chunking and Moore-Penrose generalized inverses of matrices, and also make use of the special structure of centrosymmetric matrices.
A denote transposed matrices and Moore-Penrose generalized inverses of the matrices A , n I denote real matrices of order n and,  denote the Frobenius norms of the matrices.nn CSR  denote centrosymmetric matrices, n n GCSR  represents the complete set of generalized centrosymmetric matrices of order n .
2 and Lemma 2.5, the equivalence of Problem 1 implies the existence of both