Abstract
We consider a structured inverse eigenvalue problem in which the eigenvalues of a real symmetric matrix are specified and selected entries may be constrained to take specific numerical values or to be nonzero. This includes the problem of specifying the graph of the matrix, which is determined by the locations of zero and nonzero entries. In this article, we develop a numerical method for constructing a solution to the structured inverse eigenvalue problem. The problem is recast as a constrained optimization problem over the orthogonal manifold, and a numerical optimization routine seeks its solution.
References
[1] P.-A. Absil, R. Mahony, and R. Sepulchre. Optimization algorithms on matrix manifolds. Princeton University Press, Princeton, NJ, 2008.10.1515/9781400830244Search in Google Scholar
[2] Dimitri P. Bertsekas. Constrained optimization and Lagrange multiplier methods. Computer Science and Applied Mathematics. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1982.Search in Google Scholar
[3] E. G. Birgin and J. M. Martínez. Practical augmented Lagrangian methods for constrained optimization, volume 10 of Fundamentals of Algorithms. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2014.10.1137/1.9781611973365Search in Google Scholar
[4] N. Boumal, B. Mishra, P.-A. Absil, and R. Sepulchre. Manopt, a Matlab toolbox for optimization on manifolds. Journal of Machine Learning Research, 15:1455–1459, 2014.Search in Google Scholar
[5] Moody T. Chu. Inverse eigenvalue problems. SIAM Rev., 40(1):1–39, 1998.10.1137/S0036144596303984Search in Google Scholar
[6] Alan Edelman, Tomás A. Arias, and Steven T. Smith. The geometry of algorithms with orthogonality constraints. SIAM J. Matrix Anal. Appl., 20(2):303–353, 1999.10.1137/S0895479895290954Search in Google Scholar
[7] Charles R. Johnson and Carlos M. Saiago. Eigenvalues, multiplicities and graphs, volume 211 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, 2018.10.1017/9781316155158Search in Google Scholar
[8] Charles R. Johnson, Brian D. Sutton, and Andrew J. Witt. Implicit construction of multiple eigenvalues for trees. Linear Multilinear Algebra, 57(4):409–420, 2009.10.1080/03081080701852756Search in Google Scholar
[9] Jorge Nocedal and Stephen J. Wright. Numerical optimization. Springer Series in Operations Research and Financial Engineering. Springer, New York, second edition, 2006.Search in Google Scholar
© 2019 Brian D. Sutton, published by De Gruyter
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