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 { λ 1 , … , λ p } ∈ C p × p , λ i ≠ λ j for i ≠ j , i , j = 1 , … , p , X = [ x 1 , … , x p ] ∈ C n × p , rank ( X ) = p , and both Λ and X are closed under complex conjugation in the sense that λ 2 j = λ ̄ 2 j − 1 ∈ C , x 2 j = x ̄ 2 j − 1 ∈ C n for j = 1 , … , l , and λ k ∈ R , x k ∈ R n for k = 2 l + 1 , … , p , find real-valued symmetric matrices D and K such that X Λ 2 + D X Λ + K X = 0 . Then we consider a best approximation problem: given D ̃ , K ̃ ∈ R n × n , find ( D ˆ , K ˆ ) ∈ S D K such that ‖ ( D ˆ , K ˆ ) − ( D ̃ , K ̃ ) ‖ W = min ( D , K ) ∈ S D K ‖ ( D , K ) − ( D ̃ , K ̃ ) ‖ W , where ‖ ⋅ ‖ W is a weighted Frobenius norm and S D K is the solution set of IMQEP. We show that the best approximation solution ( D ˆ , K ˆ ) is unique and derive an explicit formula for it.