二次特征值反问题的中心斜对称解及其最佳逼近

The Centroskew Symmetric Solution of the Inverse Quadratic Eigenvalue Problem and Its Optimal Approximation

利用矩阵的奇异值分解，讨论构造n阶中心斜对称矩阵M，C和K，使得二次束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SMCK的通式．然后考虑从解集SMCK中求给定矩阵[M^-，C^-，K^-]的最佳逼近问题，给出了最佳逼近解的存在唯一性及表达式．

The inverse eigenvalue problem of constructing centroskew 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 is considered by singular value decomposition of matrix. It is shown that the problem is solvable and the general expression of solution to the problem is provided. The optimal approximation problem associated with SMCK is posed, that is： to find the nearest triple matrix [M^-,C^-,K^-] from SMCK . The existence and uniqueness of the optimal approximation problem is discussed and the expression is provided for the optimal approximation problem.