矩阵方程组的双对称最小二乘解

Least-Square Bisymmetric Solutions of Matrix Equations

矩阵方程组∑_(j=1)~lA_(ij)X_jB_(ij)=C_i(i=1,2,…,t)在控制与系统领域中具有广泛应用.构造了一种算法求这个矩阵方程组的最小二乘双对称解.在没有舍入误差的情况下,该算法在有限步内能求得双对称矩阵组[X_1,X_2,…,X_l],使得∑_(i=1)~t‖∑_(j=1)~lA_(ij)X_jB_(ij)-C_i‖2达到最小.数值实验表明,这种算法是有效的.

The matrix equations ∑_（j=1）~lA_（ij）X_jB_（ij）=C_i（ i = 1,2,…,t） have numerous applications in control and system theory. An algorithm is constructed to find the least-square bisymmetric solutions of the matrix equations in this paper. The algorithm produces suitable[X1,X2,…,Xl]such that ∑_（i=1）~t‖∑_（j=1）~lA_（ij）X_jB_（ij）-C_i‖2 is minimized within a finite iteration steps in the absence of round off errors.Numerical examples are given to show that the algorithm is efficient.