The Quadratic Eigenvalue Problem
SIAM Review
| Hi-index | 7.29 |
In this paper, we first give the representation of the general solution of the following inverse monic quadratic eigenvalue problem (IMQEP): given matrices @L=diag{@l"1,...,@l"p}@?C^p^x^p, @l"i@l"j for ij, i,j=1,...,p, X=[x"1,...,x"p]@?C^n^x^p, rank(X)=p, and both @L and X are closed under complex conjugation in the sense that @l"2"j=@l@?"2"j"-"1@?C, x"2"j=x@?"2"j"-"1@?C^n for j=1,...,l, and @l"k@?R, x"k@?R^n for k=2l+1,...,p, find real-valued symmetric matrices D and K such that X@L^2+DX@L+KX=0. Then we consider a best approximation problem: given D@?,K@?@?R^n^x^n, find (D@?,K@?)@?S"D"K such that @?(D@?,K@?)-(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@?,K@?) is unique and derive an explicit formula for it.