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

Wei-Ru Xu, Qian-Yu Shu and Natália Bebiano

Applied Mathematics and Computation, 2025, vol. 488, issue C

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.

Keywords: Inverse eigenvalue problem; Pseudo-Jacobi matrix; Rank-one modification; Indefinite inner product; Spectrum distribution (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:488:y:2025:i:c:s0096300324005794

DOI: 10.1016/j.amc.2024.129118

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