Generalized Inverse Eigenvalue Problems for Symmetric Arrow-head Matrices
Authors: Yongxin Yuan
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.
Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1332596Procedia APA BibTeX Chicago EndNote Harvard JSON MLA RIS XML ISO 690 PDF Downloads 1459
 A. M. Lietuofu, The Stability of the Nonlinear Adjustment Systems, Science Press, 1959 (in Chinese).
 G. W. Bing, Introduction to the Nonlinear Control Systems, Science Press, 1988 (in Chinese).
 D. P. Oleary, G. W. Stewart, Computing the eigenvalues and eigenvectors of symmetric arrowhead matrices, J. Comp. Phys. 90 (1990) 497-505.
 O. Walter, L. S. Cederbaum, J. Schirmer, The eigenvalue problem for ÔÇÿarrow- matrices, J. Math. Phys. 25 (1984) 729-737.
 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.
 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.
 A. Ben-Israel, T. N. E. Greville, Generalized inverse: theory and applications, Wiley, New York, 1974.
 P. Lancaster, M. Tismenetsky, The theory of matrices, 2nd ed., Academic Press, New York, 1985.
 J. P. Aubin, Applied Functional Analysis, John Wiley & Sons, Inc, 1979.