A pseudo-Jacobi inverse eigenvalue problem with a rank-one modification

Abstract

Let H=diag(δ1,δ2,…,δn) be a signature matrix, where δk∈{−1,+1}. Consider Rn endowed with the indefinite inner product 〈x,y〉H:=〈Hx,y〉=yTHx for all x,y∈Rn. A pseudo-Jacobi matrix of order n is a real tridiagonal symmetric matrix with respect to this indefinite inner product. In this paper, the reconstruction of a pair of n-by-n pseudo-Jacobi matrices, one obtained from the other by a rank-one modification, is investigated given their prescribed spectra. Necessary and sufficient conditions under which this problem has a solution are presented. As a special case of the obtained results, a related problem for Jacobi matrices proposed by de Boor and Golub is thoroughly solved.