On a class of inverse quadratic eigenvalue problem

In this paper, we first give the representation of the general solution of the following inverse monic quadratic eigenvalue problem (IMQEP): given matrices Λ=diag{λ1,…,λp}∈Cp×p, λi≠λj for i≠j, i,j=1,…,p, X=[x1,…,xp]∈Cn×p, rank(X)=p, and both Λ and X are closed under complex conjugation in the sense that λ2j=λ̄2j−1∈C, x2j=x̄2j−1∈Cn for j=1,…,l, and λk∈R, xk∈Rn for k=2l+1,…,p, find real-valued symmetric matrices D and K such that XΛ2+DXΛ+KX=0. Then we consider a best approximation problem: given D̃,K̃∈Rn×n, find (Dˆ,Kˆ)∈SDK such that ‖(Dˆ,Kˆ)−(D̃,K̃)‖W=min(D,K)∈SDK‖(D,K)−(D̃,K̃)‖W, where ‖⋅‖W is a weighted Frobenius norm and SDK is the solution set of IMQEP. We show that the best approximation solution (Dˆ,Kˆ) is unique and derive an explicit formula for it.