基于正交投影方法的二次特征值反问题及其最佳逼近解

摘要

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.%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.