Sufficient conditions for the solubility of inverse eigenvalue problems

This is a sequel to our previous paper [4] where we initiated a study of inverse eigenvalue problems for matrices in the multiparameter setting. The one parameter version of the problem under consideration asks for conditions on a given n × n symmetric matrix A and on n given real numbers s1≦s2≦…≦sn under which a diagonal matrix V can be found so that A + V has sl,…,sn as its eigenvalues. Our motivation for this problem and our method of attack on it in [4]p comes chiefly from the work of Hadeler [5] in which sufficient conditions were given for existence of the desired diagonal V. Hadeler's approach in [5] relied heavily on the Brouwer fixed point theorem and this was also our main tool in [4]. Subsequently, using properties of topological degree, Hadeler [6] gave somewhat different conditions for the existence of the diagonal V. It is our desire here to follow this lead and to use degree theory to give some results extending those in [6] to the multiparameter case.

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Some sufficient conditions for the solvability of inverse eigenvalue problems

Linear Algebra and its Applications ◽

10.1016/0024-3795(91)90095-e ◽

1991 ◽

Vol 148 ◽

pp. 225-236 ◽

Cited By ~ 5

Author(s):

Luoluo Li

Keyword(s):

Eigenvalue Problems ◽

Sufficient Conditions ◽

Inverse Eigenvalue Problems ◽

Inverse Eigenvalue

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On inverse eigenvalue problems for two kinds of special banded matrices

Filomat ◽

10.2298/fil1702371x ◽

2017 ◽

Vol 31 (2) ◽

pp. 371-385 ◽

Cited By ~ 4

Author(s):

Wei-Ru Xu ◽

Guo-Liang Chen

Keyword(s):

Eigenvalue Problems ◽

Sufficient Conditions ◽

Numerical Algorithms ◽

Tridiagonal Matrices ◽

Inverse Eigenvalue Problems ◽

Banded Matrices ◽

Inverse Eigenvalue ◽

Maximal Eigenvalues ◽

Principal Submatrices ◽

Necessary And Sufficient

This paper presents two kinds of symmetric tridiagonal plus paw form (hereafter TPPF) matrices, which are the combination of tridiagonal matrices and bordered diagonal matrices. In particular, we exploit the interlacing properties of their eigenvalues. On this basis, the inverse eigenvalue problems for the two kinds of symmetric TPPF matrices are to construct these matrices from the minimal and the maximal eigenvalues of all their leading principal submatrices respectively. The necessary and sufficient conditions for the solvability of the problems are derived. Finally, numerical algorithms and some examples of the results developed here are given.

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A Generalized Inexact Newton Method for Inverse Eigenvalue Problems

Abstract and Applied Analysis ◽

10.1155/2014/721346 ◽

2014 ◽

Vol 2014 ◽

pp. 1-6 ◽

Cited By ~ 1

Author(s):

Weiping Shen

Keyword(s):

Newton Method ◽

Quadratic Convergence ◽

Eigenvalue Problems ◽

Generalized Newton Method ◽

Inexact Newton Method ◽

Numerical Tests ◽

Inverse Eigenvalue Problems ◽

Inverse Eigenvalue ◽

Generalized Newton ◽

Special Case

We propose a generalized inexact Newton method for solving the inverse eigenvalue problems, which includes the generalized Newton method as a special case. Under the nonsingularity assumption of the Jacobian matrices at the solutionc*, a convergence analysis covering both the distinct and multiple eigenvalue cases is provided and the quadratic convergence property is proved. Moreover, numerical tests are given in the last section and comparisons with the generalized Newton method are made.

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