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

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. •In this paper, we propose a new inverse eigenvalue problem for pseudo-Jacobi matrices, which requires us to reconstruct a pair of pseudo-Jacobi matrices from the given spectra of an n-by-n pseudo-Jacobi matrix and another one matrix, obtained by a rank-one modification.•For the special signature operator, the spectral distribution of the pseudo-Jacobi matrix and another matrix, formed by its rank-one modification, are given.•Necessary and sufficient conditions under which this problem has solution are presented. As a special case of the obtained results, a related problem for Jacobi matrices proposed by de Boor and Golub [LAA, 21 (1978) 245-260] is thoroughly solved.