A symmetric inverse eigenvalue problem in structural dynamic model updating

In this paper, we first give the representation of the general solution of the following inverse eigenvalue problem (IEP): Given X∈Rn×p and a diagonal matrix Λ∈Rp×p, find real-valued symmetric (2r+1)-diagonal matrices M and K such that MXΛ=KX. We then consider an optimal approximation problem: Given real-valued symmetric (2r+1)-diagonal matrices Ma,Ka∈Rn×n, find (Mˆ,Kˆ)∈SMK such that ‖Mˆ-Ma‖2+‖Kˆ-Ka‖2=inf(M,K)∈SMK(‖M-Ma‖2+‖K-Ka‖2), where SMK is the solution set of IEP. We show that the optimal approximation solution (Mˆ,Kˆ) is unique and derive an explicit formula for it.