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 Lambda diag{lambda(1),...,lambda(p)} is an element of C-pxp, X = [x(1),...,x(p)] is an element of C-nxp, and both Lambda and X are closed under complex conjugation in the sense that lambda(2j) = (lambda) over bar (2j-1) is an element of C, x(2j) = (x) over bar (2j-1) is an element of C-n for j = 1,..., l, and lambda(k) is an element of R, x(k) is an element of R-n for k = 2l + 1,..., p, find real-valued symmetric (2r + 1)-diagonal matrices M, D and K such that MX Lambda(2) + DX Lambda + KX = 0. We then consider an optimal approximation problem: given real- valued symmetric (2r + 1)-diagonal matrices M-a, D-a, K-a is an element of R-nxn, find ((M) over cap, (D) over cap, (K) over cap) is an element of S-E such that parallel to(M) over cap - M-a parallel to(2) + parallel to(D) over cap - D-a parallel to(2) + parallel to(K) over cap - K-a parallel to(2) = inf((M, D, K)) is an element of S-E (parallel to M - M-a parallel to(2) + parallel to D - D-a parallel to(2) + parallel to K - K-a parallel to(2)), where S-E is the solution set of IQEP. We show that the optimal approximation solution ((M) over cap, (D) over cap, (K) over cap) is unique and derive an explicit formula for it. Copyright (C) 2008 Yongxin Yuan.