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

Let $H=\text{diag}({\delta }_1,{\delta }_2,\dots ,{\delta }_n)$ be a signature matrix, where ${\delta }_k\in \{-1,+1\}$. Consider $R^n$ endowed with the indefinite inner product ${〈x,y〉}_H:=〈Hx,y〉=y^{T}Hx$ for all $x,y\in R^n$. 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.