矩阵约束下矩阵方程组的双对称最小二乘解

摘要/Abstract

摘要：

矩阵方程组 $\100dpi \inline $\sum\limits_{j=1}^lA_{ij}X_jB_{ij}=C_i\;(i=1,2,\cdots,t)$$ 在控制与系统领域中具有广泛应用. 该文构造了一种算法求解这个矩阵方程组, 其中 $\100dpi \inline $X_j\in R^{n_j \times n_j}\;(j=1,2,\cdots,l)$$ 为带有特殊中心主子矩阵约束的双对称矩阵. 在没有舍入误差的情况下, 该算法经过有限步迭代得到 $\100dpi \inline $[X_1,X_2,\cdots,X_l]$$ , 使得 $\100dpi \inline $\sum\limits_{i=1}^t\|\sum\limits_{j=1}^lA_{ij}X_jB_{ij}-C_i\|=\min$$ . 实例表明这种方法是有效的.

关键词: 矩阵方程组 , 中心主子矩阵 , 双对称解 , 子矩阵约束 , 最小二乘解.

Abstract:

The general coupled matrix equations $\100dpi \inline $\sum\limits_{j=1}^lA_{ij}X_jB_{ij}=C_i,\;\;i=1,2,\cdots,t$$ ,   have numerous applications in control and system theory.    In this paper, an algorithm is constructed   to solve the   general coupled matrix equations where   $\100dpi \inline $X_j\in R^{n_j\times n_j}\;(j=1,2,\cdots,l)$$ is a bisymmetric  matrix with a specified central  principal submatrix. The algorithm produces suitable
   $\100dpi \inline $[X_1,X_2,\cdots,X_l]$$ such that    $\100dpi \inline $\sum\limits_{i=1}^t\|\sum\limits_{j=1}^lA_{ij}X_jB_{ij}-C_i\|$$    is minimized within a finite iteration steps in the absence of roundoff errors.
The algorithm requires little storage capacity.    Numerical examples are given to show that the algorithm is efficient.

Key words: General coupled matrix equations , Central principal submatrix , Bisymmetric solution , Submatrix constraint , Least squares solution.

中图分类号:

65F10

彭卓华. 矩阵约束下矩阵方程组的双对称最小二乘解[J]. 数学物理学报, 2015, 35(1): 131-150.

PENG Zhuo-Hua. The Bisymmetric Least Squares Solutions of the General Coupled |Matrix Equations with A Submatrix Constraint[J]. Acta mathematica scientia,Series A, 2015, 35(1): 131-150.

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