对称矩阵反问题的总体最小二乘解

The Total Least Squares Solution for the Inverse Problems of Symmetric Matrices

最小二乘法是近年来求解对称矩阵反问题的一种常用方法,但因系数矩阵常常存在误差,方法本身具有很大的局限性。鉴于此,本文提出并讨论了对称矩阵反问题的总体最小二乘解,给出了解的一般表达式;证明了最佳逼近问题解的存在唯一性,给出了其具体表达式及数值算法,并将数值结果应用于求解对称矩阵反问题。

Least squares have been widely used in the inverse problems of symmetric matrices. However, errors always occur in the relevant coefficient matrix. This makes the approach limited. In order to overcome this shortcoming, the total least squares solution of inverse problems of symmetric matrices are proposed. The general form of the solution is given. The existence and expression of the optimal approximation solution are presented, and a numerical algorithm is derived. These results are finally applied to solve the inverse problem of symmetric matrices.