Abstract:

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)=λ²M+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 a
ssociated with S_MCK was posed, that is, to find the nearest triple matrix  from S_MCK. The existence and uniqueness of the optimal approximation problem was discussed and the exoression was provided for the optimal approximation problem.

Key words: quadratic eigenvalue problem, symmetric and skew antisymmetric matrix; , inverse problem, optimal approximation, singular value decomposition