二次特征值反问题的对称次反对称解及其最佳逼近

摘要

利用矩阵的奇异值分解和矩阵的Kronecker乘积, 讨论构造对称次反对称矩阵M,C和K, 使得二次约束Q(λ)=λ~2M+λC+K具有给定特征值和特征向量的特征值反问题. 首先证明反问题是可解的, 并给出了解集S_(MCK)的通式. 进而考虑了解集S_(MCK)中对给定矩阵((M),(C),(K))的最佳逼近问题, 得到了最佳逼近解.%The inverse eigenvalue problem of constructing symmetric and skew anti-symmetric matrices M,C and K of size n for the quadratic pencil Q(λ) = λ~2M + λC + K so that Q(λ) has a prescribed subset of eigenvalues and eigenvectors was considered by means of singular value decomposition of matrix and Kronecker product of matrices. The problem was firstly improved to be solvable and the general expression of the solution to the problem was provided. The optimal approximation problem associated with S_(MCK) was posed, that is, to find the nearest triple matrix ((M),(C),(K)) from S_(MCK). The existence and uniqueness of the optimal approximation problem was discussed and the exoression was provided for the optimal approximation problem.