Inverse eigenvalue problem for bordered diagonal matrices

Subject Areas : Numerical analysis

S. Mashayekhi ¹ (Department of Mathematics, Faculty of Sciences, Arak University, Arak 38156-8-8349, Iran)
S. M. Karbassi ² (Faculty of Mathematics, Yazd University, Yazd, Iran)
S. A. Shahzadefazeli ³ (Faculty of Mathematics, Yazd University, Yazd, Iran)

Received: 2021-08-27 Accepted : 2021-11-14 Published : 2021-12-01

Keywords: Inverse eigenvalue problem, bordered diagonal matrix, leading principal submatrix, minimal eigenvalue, maximal eigenvalue,

Abstract :

‎In this paper, the inverse eigenvalue problem for the bordered diagonal matrices are reconsidered whose elements are equal to zero except for the first row, the first column and the diagonal elements. The necessary and sufficient conditions for existence of a symmetric bordered diagonal matrix from special spectral data have been determined. A new algorithm to make such matrices is derived and some numerical examples are given to illustrate the efficiency of the method.

References:

[1] G. W. Bing, Introduction to the Nonlinear Control Systems, Science Press, 1988 (in Chinese).

[2] M. T. Chu, G. H. Golub, Inverse Eigenvalue Problems: Theory, Algorithms and applications, Oxford University Press, New York, 2005.

[3] L. Elsner, C. He, Perturbation and interlace theorems for the unitary eigenvalue problem, Linear Algebra and its Applications. 188 (1993), 207-229. 336-347.

[4] G. M. L. Gladwell, N. B. Willms, The construction of tridiagonal system from its frequency response at an interior point, Inverse Problems 4 (1988), 1013-1024.

[5] A. M. Lietuofu, The Stability of the Nonlinear Adjustment Systems, Science Press, 1959 (in Chinese).

[6] J. Peng, X. Y. Hu, L. Zhang, Two inverse eigenvalue problems for a special kind of matrices, Linear Algebra and its Applications. 416 (2006), 336-347.

[7] H. Pickman, J. Egana, R. L. Soto, Extermal inverse eigenvalue problem for bordered diagonal matrices, Linear Algebra and its Applications. 427 (2007), 256-271.