Iterative Methods For Constrained Matrix Equation AXB+CYD=E And Associated Optimal Approximation

To find solution of matrix equation in some given matrix sets is so-called the constrainedequation problem. In this paper, we introduce the concepts of generalized centrosymmetricmatrix. Whereafter, we establish the iterative algorithms for solving the generalizedcentro-symmetric solution of the matrix equation AXB + CYD = E and the generalized solution of its least-square solution. Meanwhile, when the matrix equation is consistent, for given matrix pair [X₀, Y₀], we obtain the optimal approximation solution in the solution set of the matrix equation problems. By the associated iterative method, We show that, for arbitrary initial iterative matrix pair [X₁, Y₁], the solution of matrix equation can be obtained within finite iterative steps in the absence of roundoff errors. Especially, if let X₁ = 0, Y₁ = 0 or X₁,Y₁ be some particular matrix, the solution obtainedby the iterative algorithm is the least Frobenius norm solution. In addition, by the iterative algorithm, we can represent the associated optimal approximation solution pair [X, Y], which can be derived by the least-norm solution [X*, Y*] of the new matrix equationAXB + CYD = E, where X = X - X₀, Y = Y - Y₀, E = E - AX₀B - CY₀D, i.e., X = X* + X₀, Y = Y* + Y₀. Finally, we give associated numerical examples, which illustrate the efficiency of the iterative methods.