Commenced in January 2007
Frequency: Monthly
Edition: International
Paper Count: 30172
Generalized Inverse Eigenvalue Problems for Symmetric Arrow-head Matrices

Authors: Yongxin Yuan

Abstract:

In this paper, we first give the representation of the general solution of the following inverse eigenvalue problem (IEP): Given X ∈ Rn×p and a diagonal matrix Λ ∈ Rp×p, find nontrivial real-valued symmetric arrow-head matrices A and B such that AXΛ = BX. We then consider an optimal approximation problem: Given real-valued symmetric arrow-head matrices A, ˜ B˜ ∈ Rn×n, find (A, ˆ Bˆ) ∈ SE such that Aˆ − A˜2 + Bˆ − B˜2 = min(A,B)∈SE (A−A˜2 +B −B˜2), where SE is the solution set of IEP. We show that the optimal approximation solution (A, ˆ Bˆ) is unique and derive an explicit formula for it.

Keywords: Partially prescribed spectral information, symmetric arrow-head matrix, inverse problem, optimal approximation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1332596

Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1376

References:


[1] A. M. Lietuofu, The Stability of the Nonlinear Adjustment Systems, Science Press, 1959 (in Chinese).
[2] G. W. Bing, Introduction to the Nonlinear Control Systems, Science Press, 1988 (in Chinese).
[3] D. P. Oleary, G. W. Stewart, Computing the eigenvalues and eigenvectors of symmetric arrowhead matrices, J. Comp. Phys. 90 (1990) 497-505.
[4] O. Walter, L. S. Cederbaum, J. Schirmer, The eigenvalue problem for ÔÇÿarrow- matrices, J. Math. Phys. 25 (1984) 729-737.
[5] 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.
[6] C. F. Borges, R. Frezza, W. B. Gragg, Some inverse eigenproblems for Jacobi and arrow matrices, Numerical Linear Algebra with Applications. 2 (1995) 195-203.
[7] A. Ben-Israel, T. N. E. Greville, Generalized inverse: theory and applications, Wiley, New York, 1974.
[8] P. Lancaster, M. Tismenetsky, The theory of matrices, 2nd ed., Academic Press, New York, 1985.
[9] J. P. Aubin, Applied Functional Analysis, John Wiley & Sons, Inc, 1979.