Generalized inverse spectral problem for pseudo-Jacobi matrices with mixed eigendata

In this paper, we investigate a generalized inverse eigenproblem for pseudo-Jacobi matrices with mixed eigendata. These matrices appear in non-Hermitian Quantum Mechanics and extend the well-known concept of Jacobi matrices. It is shown that a unique pseudo-Jacobi matrix may be recovered from certain prescribed mixed eigendata, i.e. its leading principal submatrix, two distinct real eigenvalues, and part of the corresponding eigenvectors. An algorithm is provided for the reconstruction of such a matrix, and illustrative numerical experiments are presented to test the algorithm. The recurrence relation involving leading principal minors is crucial for the problem solution.