The Solution For Two Kinds Constrained Matrix Equation Problems And Associated Optimal Approximation

To find solution of matrix equation in some given matrix sets is so-called the constrained matrix problem. In this paper, we first introduce the concepts and structures of (R, S)-symmetric-matrices, (R, S)-skew-symmetric matrices. Whereafter, we derive the (R, S)-skew-symmetric solution for the matrix equation AXB = C by using generalized singular-value decomposition; We get the (R,S)-symmetric solution for the matrix equation AXB + CYD = E by establishing an efficient algorithm. Meanwhile, the optimal unique approximation is considered.According to the properties of (R, S)-skew-symmetric matrices, we derive the necessary and sufficient conditions and expression for (R, S)-skew-symmetric solution of matrix equation AXB = C. Moreover, for a arbitrary given matrix X^*∈C^m×n, the optimal unique approximation (?)∈S_E and its expression is provided. For the iterative method of the matrix equation AXB + CYD = E, we prove 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 and the solvability of the matrix equation can be determined automatically in the iterative process. Especially, if let X₁ = 0, Y₁ = 0 or X₁, Y₁ have some particular form, 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 pair [(?), (?)] for any arbitrary given matrix pair [X^*, Y^*], which can be derived by the least-norm solution [(?)^*, (?)^*] of the new matrix equation A(?)B + C(?)D = (?), where (?)= X-X^*, (?) = Y-Y^*,(?) = E-AX^*B-CY^*D, i.e.,(?) = (?)^*+X^*, (?) = (?)^*+Y^*. Finally, we give associated numerical examples, which illustrate the efficiency of the iterative methods.