Iterative Methods For Several Constrained Matrix Equation Problems And Associated Optimal Approximation

To find solutions of matrix equation in some given matrix sets is so-called the constrained matrix equation problem. In this paper, we first introduce the concepts and structures of generalized per-symmetric matrix, generalized centro-symmetric matrix or generalized bisymmetric matrix. Whereafter, we establish the iterative algorithms for solving the constrained matrix equation AXB = C and matrix equations A₁XB₁ = C₁, A₂XB₂ = C₂ over unknown X, where the matrix X is generalized per-symmetric, generalized centro-symmetric or generalized bisymmetric. Meanwhile, when the matrix equation(s) is consistent, for given matrix X₀, 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 X₁, the solution of matrix equation(s) can be obtained within finite iterative steps in the absence of roundoff errors. Especially, if let X-1 = 0 or X₁ be some particular matrix, the solution obtained by the iterative algorithm is the least Frobenius norm solution. In addition, by the iterative algorithm, we can represent the associated optimal approximation solution X|^, which can be derived by the least-norm solution X* of the new matrix equation AX|^B = C| or matrix equations A₁X|^B₁ = (C|⁾₁, A₂X|^B₂ = (C|⁾₂, where X| = X—X₀, C| = C—AX₀B, (C|⁾_i = C_i—A_iX₀B_i (i = 1,2), i.e., X|^ = X|^* + X₎. Finally, we give associated numerical examples, which illustrate the efficiency of the iterative methods.