反自反矩阵的二次特征值反问题及其最佳逼近

二次特征值反问题是二次特征值问题的一个逆过程，在结构动力模型修正领域中应用非常广泛．本文由给定的部分特征值和特征向量，利用矩阵分块法、奇异值分解和Moore-Penrose广义逆，分析了二次特征值反问题反自反解的存在性，得出了解的一般表达式．然后讨论了任意给定矩阵在解集中最佳逼近解的存在性和唯一性．最后给出解的表达式和数值算法，由算例验证了结果的正确性．

The Anti-reflexive Solution of the Inverse Quadratic Eigenvalue Problem and Its Optimal Approximation

The inverse problem of quadratic eigenvalue is an inverse process of quadratic eigenvalue problem, and it is widely used in the field of structural dynamic model correction. Given part of eigenvalues and eigenvectors, based on the singular value decomposition of matrix, block matrix method and generalized inverse of Moore-Penrose, the inverse quadratic eigenvalue problem of constructing anti-reflexive matrices is considered in this paper. Then, a general expression of solution to the problem is presented. Moreover, the existence and uniqueness of the optimal approximation problem associated with solution set is discussed. Finally, the expression and numerical method are proposed, the correctness of the result is verified by a numerical example.