An algorithm for constructing a pseudo‐Jacobi matrix from given spectral data. Issue 2 (18th September 2012)

• Record Type: Journal Article
• Authors: Bebiano, N.; Furtado, S.; da Providência, J.
• Other Names: Chu Eric guestEditor.; Lin Wen‐wei guestEditor.; Reichel Lothar guestEditor.
• Is Part Of: Numerical linear algebra with applications. Volume 20:Issue 2(2013:Mar.)
• Page Start: 185
• Page End: 197
• Publication Date: 2012-09-18
• Subjects: Indefinite inner product -- inverse eigenvalue problem -- Jacobi matrix -- pseudo‐Jacobi matrix; Algebras, Linear -- Periodicals; 512.5
• DOI: 10.1002/nla.1855
• Languages: English
• ISSNs: 1070-5325

Abstract

SUMMARY: The main purpose of this paper is the extension of the classical spectral direct and inverse analysis of Jacobi matrices for the non‐self‐adjoint setting. Matrices of this class appear in the context of non‐Hermitian quantum mechanics. The reconstruction of a pseudo‐Jacobi matrix from its spectrum and the spectra of two complementary principal matrices is investigated in the context of indefinite inner product spaces. An existence and uniqueness theorem is given, and a strikingly simple algorithm, based on the Euclidean division algorithm, to reconstruct the matrix from the spectral data is presented. A result of Friedland and Melkman stating a necessary and sufficient condition for a real sequence to be the spectrum of a non‐negative Jacobi matrix is revisited and generalized. Namely, it is shown that a suitable set of prescribed eigenvalues defines a unique non‐negative pseudo‐Jacobi matrix, which is J ‐Hermitian for a fixed J . Copyright © 2012 John Wiley & Sons, Ltd.