Yıl 2019, Cilt 68, Sayı 2, Sayfalar 1273 - 1288 2019-07-01

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

Ebru Ergun [1]

10 36

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.
Jacobi matrix, eigenvalue, normalizing number, dissipative, inverse spectral problem
  • 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.
Birincil Dil en
Konular Matematik
Dergi Bölümü Makaleler
Yazarlar

Orcid: 0000-0003-4873-6191
Yazar: Ebru Ergun (Sorumlu Yazar)
Kurum: Department of Mathematics, Karabuk University.
Ülke: Turkey


Bibtex @araştırma makalesi { cfsuasmas419098, journal = {Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics}, issn = {1303-5991}, eissn = {2618-6470}, address = {Ankara Üniversitesi}, year = {2019}, volume = {68}, pages = {1273 - 1288}, doi = {10.31801/cfsuasmas.419098}, title = {On the inverse problem for finite dissipative Jacobi matrices with a rank-one imaginary part}, key = {cite}, author = {Ergun, Ebru} }
APA Ergun, E . (2019). On the inverse problem for finite dissipative Jacobi matrices with a rank-one imaginary part. Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics, 68 (2), 1273-1288. DOI: 10.31801/cfsuasmas.419098
MLA Ergun, E . "On the inverse problem for finite dissipative Jacobi matrices with a rank-one imaginary part". Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 (2019): 1273-1288 <http://dergipark.gov.tr/cfsuasmas/issue/42777/419098>
Chicago Ergun, E . "On the inverse problem for finite dissipative Jacobi matrices with a rank-one imaginary part". Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 (2019): 1273-1288
RIS TY - JOUR T1 - On the inverse problem for finite dissipative Jacobi matrices with a rank-one imaginary part AU - Ebru Ergun Y1 - 2019 PY - 2019 N1 - doi: 10.31801/cfsuasmas.419098 DO - 10.31801/cfsuasmas.419098 T2 - Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics JF - Journal JO - JOR SP - 1273 EP - 1288 VL - 68 IS - 2 SN - 1303-5991-2618-6470 M3 - doi: 10.31801/cfsuasmas.419098 UR - http://dx.doi.org/10.31801/cfsuasmas.419098 Y2 - 2018 ER -
EndNote %0 Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics On the inverse problem for finite dissipative Jacobi matrices with a rank-one imaginary part %A Ebru Ergun %T On the inverse problem for finite dissipative Jacobi matrices with a rank-one imaginary part %D 2019 %J Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics %P 1303-5991-2618-6470 %V 68 %N 2 %R doi: 10.31801/cfsuasmas.419098 %U 10.31801/cfsuasmas.419098
ISNAD Ergun, Ebru . "On the inverse problem for finite dissipative Jacobi matrices with a rank-one imaginary part". Communications Faculty of Sciences University of Ankara Series A1 Mathematics and Statistics 68 / 2 (Temmuz 2019): 1273-1288. http://dx.doi.org/10.31801/cfsuasmas.419098