An inverse eigenvalue problem for Jacobi matrices with a missing eigenvalue

We consider an inverse eigenvalue problem for constructing an n x n Jacobi matrix J(n) under the circumstance that its all eigenvalues, except for one and a part of the matrix Jn are given. To be precise, the known partial data of J(n) means either its leading principal submatrix J([(n+1)/2]) when n is odd, or the submatrix J([(n+1)/2]) together with the [(n + 1)/2] x (n/2 + 1) codiagonal element when n is even. The necessary and sufficient conditions for the solvability of the problem is derived, also the numerical algorithm and a numerical example are provided. (C) 2022 Elsevier Ltd. All rights reserved.

Automated summary

This paper discusses the inverse eigenvalue problem of constructing a Jacobi matrix under the given eigenvalues and partial matrix data. The necessary and sufficient conditions for the solvability of the problem are derived, and a numerical algorithm and example are provided.