基于正交投影方法的二次特征值反问题及其最佳逼近解

Inverse Quadratic Eigenvalue Problem and Its Optimal Approximation Solution Based on Orthogonal Projection Methods

考虑二次特征值反问题的广义中心对称解(广义反中心对称解)及其最佳逼近问题,应用矩阵的正交投影方法,给出矩阵方程AX+BY+CZ=0的解及其最佳逼近问题.利用广义中心对称矩阵(广义反中心对称矩阵)的性质导出了该问题有广义中心对称解(广义反中心对称解)的条件及有解情况下的通解表达式,并证明了最佳逼近问题解的存在性与唯一性,得到了最佳逼近解的表达式.

We considered the generalized centrosymmetric solution （generalized anti-centrosymmetric solution） of an inverse quadratic eigenvalue problem and its optimal approximation problem. By using the orthogonal projection methods of matrix, we gave the solution of matrix equation AX＋BY＋CZ= 0 and its optimal approximation problem. According to the properties of generalized centrosymmetric matrices （generalized anti-centrosymmetric matrices）, we derived the conditions for the problem with a generalized centrosymmetric solution （generalized anti-centrosymmetric solution） and the expression of general solution. We proved the existence and the uniqueness of solution of the optimal approximation problem, and obtained the expression of the optimal approximation solution.