An inverse quadratic eigenvalue problem for damped structural systems.

We first give the representation of the general solution of the following inverse quadratic eigenvalue problem IQEP : given Λ diag{λ1, . . . , λp} ∈ Cp×p , X x1, . . . , xp ∈ Cn×p, and both Λ and X are closed under complex conjugation in the sense that λ2j λ2j−1 ∈ C, x2j x2j−1 ∈ C for j 1, . . . , l, and λk ∈ R, xk ∈ R for k 2l 1, . . . , p, find real-valued symmetric 2r 1 -diagonal matrices M, D and K such thatMXΛ DXΛ KX 0. We then consider an optimal approximation problem: given real-valued symmetric 2r 1 -diagonal matricesMa,Da,Ka ∈ Rn×n, find M, D, K ∈ SE such that ‖ M −Ma‖ 2 ‖ D −Da‖ 2 ‖ K −Ka‖ 2 inf M,D,K ∈SE ‖M −Ma‖ ‖D −Da‖ ‖K −Ka‖ , where SE is the solution set of IQEP. We show that the optimal approximation solution M, D, K is unique and derive an explicit formula for it.