On the inverse problem for finite dissipative Jacobi matrices with a rank-one imaginary part

Year 2019, , 1273 - 1288, 01.08.2019

Ebru Ergun

https://doi.org/10.31801/cfsuasmas.419098

Cited By: 2

Abstract

This paper deals with the inverse spectral problem consisting in the reconstruction of a finite dissipative Jacobi matrix with a rank-one imaginary part from its eigenvalues. Necessary and sufficient conditions are formulated for a prescribed collection of complex numbers to be the spectrum of a finite dissipative Jacobi matrix with a rank-one imaginary part. Uniqueness of the matrix having prescribed eigenvalues is shown and an algorithm for reconstruction of the matrix from prescribed eigenvalues is given.

Keywords

Jacobi matrix, eigenvalue, normalizing number, dissipative, inverse spectral problem

References

• Arlinskii, Yu. and Tsekanovskii, E., Non-self-adjoint Jacobi matrices with a rank-one imaginary part, J. Funct. Anal., 241 (2006), 383--438.
• Boley, D. and Golub, G. H., A survey of matrix inverse eigenvalue problems, Inverse Problems, 3 (1987), 595--622.
• de Boor, C. and Golub, G. H., The numerically stable reconstruction of a Jacobi matrix from spectral data, Linear Algebra Appl., 21 (1978), 245--260.
• Fuhrmann, P. A., A Polynomial Approach to Linear Algebra, Second Edition, Springer, New York, 2012.
• Gelfand, I. M. and Levitan, B. M., On the determination of a differential equation from its spectral function, Izv. Akad. Nauk, Ser. Mat., 15 (1951), 309--360 (Russian); Engl. transl., Amer. Math. Soc. Transl., (2) 1 (1955), 253--304.
• Gesztesy, F. and Simon, B., M-functions and inverse spectral analysis for finite and semi-infinite Jacobi matrices, J. Anal. Math ., 73 (1997), 267--297.
• Gray, L. J. and Wilson, D. G., Construction of a Jacobi matrix from spectral data, Linear Algebra Appl., 14 (1976), 131--134.
• Guseinov, G. Sh., Construction of a complex Jacobi matrix from two-spectra, Hacettepe J. Math. Stat., 40 (2011), 297--303.
• Guseinov, G. Sh., On an inverse problem for two spectra of finite Jacobi matrices, Appl. Math. Comput., 218 (2012), 7573--7589. Guseinov, G. Sh., On a discrete inverse problem for two spectra, Discrete Dynamics in Nature and Society, 2012 (2012), Article ID 956407, 14 pages.
• Hald, O. H., Inverse eigenvalue problems for Jacobi matrices, Linear Algebra Appl., 14 (1976), 63--85.
• Hochstadt, H., On some inverse problems in matrix theory, Arch. Math., 18 (1967), 201--207.
• Hochstadt, H., On construction of a Jacobi matrix from spectral data, Linear Algebra Appl., 8 (1974), 435--446. Huseynov, A. and Guseinov, G. Sh., Solution of the finite complex Toda lattice by the method of inverse spectral problem, Appl. Math. Comput., 219 (2013), 5550--5563.
• Teschl, G., Jacobi Operators and Completely Integrable Nonlinear Lattices, vol. 72 of Mathematical Surveys and Monographs, American Mathematical Society, 2000.
• Ergun, E. and Huseynov, A., On an Inverse Problem for a Quadratic Eigenvalue Problem, Int. J. Difference Equ., vol. 12, 1 (2017), 13--26.