高翠金^*， 赵丽君， 胡锡炎

（  湖南大学数学与计量经济学院；  ）

摘要： 本文主要讨论了行对称矩阵的方程组问题及其最佳逼近问题.在该问题的研究过程中，我们充分利用了行对称矩阵的对称特性，对矩阵进行分块和降阶，使得求解行对称矩阵方程组问题及其最佳逼近的过程大为简化. 在本文中，通过研究行对称矩阵的基本性质，得到了行对称矩阵基本结构表达式. 在此基础上，将行对称矩阵方程组问题转化为一般的矩阵方程组问题，转化后的一般矩阵方程组问题的规模是原行对称矩阵方程组问题的一半. 然后再利用矩阵的奇异值分解，得到了行对称矩阵方程组有解的充要条件及其通解表达式；最后，根据 Frobenius 范数的正交不变性，得到了行对称矩阵方程组问题的最佳逼近解的表达式.
关键词： 行对称矩阵, 奇异值分解,最佳逼近.

Cuijin GAO^*， Lijun ZHAO， Xiyan HU

（  College of Mathematics and Econometrics, Hunan University；  ）

Abstract： In this paper, a matrix equations problem for row symmetric matrices and its optimal approximate problem are considered. In the research process of these problems, we use the symmetric property of row symmetric matrices fully and by partitioning of matrix and reduction of order,which result in considerable simplification of solving the matrix equations problem for row symmetric matrices and its optimal approximation. In this paper,first of all, we obtain the expression of basic structure of row symmetric matrices according to the properties of row symmetric matrices. Based on this, we convert the matrix equations problem for row symmetric matrices into ordinary matrix equations problem,the scale of converted matrix equations problem is half of that for the original problem. By the singular value decomposition of matrix,we obtain the necessary and sufficient condition for the existence of the solutions to matrix equations problem for row symmetric matrices, and the expression of general solutions is provided. Finally, based on the orthogonal invariance of Frobenius norm,we get the expression of the optimal approximate solution to matrix equations problem for row symmetric matrices.

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