Iterative solution to a system of matrix equations
Summary: An efficient iterative algorithm is presented to solve a system of linear matrix equations \(A_1X_1B_1+A_2X_2B_2=E\), \(C_1X_1D_1+C_2X_2D_2=F\) with real matrices \(X_1\) and \(X_2\). By this iterative algorithm, the solvability of the system can be determined automatically. When the system is consistent, for any initial matrices \(X^0_1\) and \(X^0_2\), a solution can be obtained in the absence of roundoff errors, and the least norm solution can be obtained by choosing a special kind of initial matrix. In addition, the unique optimal approximation solutions \(\hat{X}_1\) and \(\hat{X}_2\) to the given matrices \(\tilde{X}_1\) and \(\tilde{X}_2\) in Frobenius norm can be obtained by finding the least norm solution of a new pair of matrix equations \(A_1\bar{X}_1B_1+A_2\bar{X}_2B_2=\bar{E}\), \(C_1\bar{X}_1D_1+C_2\bar{X}_2D_2=\bar{F}\), where \(\bar{E}=E-A_1\tilde{X}_1B_1-A_2\tilde{X}_2B_2\), \(\bar{F}=F-C_1\tilde{X}_1D_1-C_2\tilde{X}_2D_2\). The given numerical example demonstrates that the iterative algorithm is efficient. Especially, when the numbers of the parameter matrices \(A_1,A_2,B_1,B_2,C_1,C_2,D_1,D_2\) are large, our algorithm is efficient as well.
scientific article; zbMATH DE number 6306035
scientific article