×

A test matrix for an inverse eigenvalue problem. (English) Zbl 1442.15012

Summary: We present a real symmetric tridiagonal matrix of order \(n\) whose eigenvalues are \(\{2 k \}_{k = 0}^{n - 1}\) which also satisfies the additional condition that its leading principle submatrix has a uniformly interlaced spectrum, \(\{2 l + 1 \}_{l = 0}^{n - 2}\). The matrix entries are explicit functions of the size \(n\), and so the matrix can be used as a test matrix for eigenproblems, both forward and inverse. An explicit solution of a spring-mass inverse problem incorporating the test matrix is provided.

MSC:

15A18 Eigenvalues, singular values, and eigenvectors
15B57 Hermitian, skew-Hermitian, and related matrices
PDF BibTeX XML Cite
Full Text: DOI arXiv

References:

[1] Hochstadt, H., On some inverse problems in matrix theory, Archiv der Mathematik, 18, 201-207 (1967) · Zbl 0147.27701
[2] Gladwell, G. M. L., Inverse Problems in Vibration. Inverse Problems in Vibration, Monographs and Textbooks on Mechanics of Solids and Fluids: Mechanics. Dynamical Systems, 9, x+263 (1986), Dordrecht, The Netherlands: Martinus Nijhoff Publishers, Dordrecht, The Netherlands
[3] Brad Willms, N., Some matrix inverse eigenvalue problems [M.S. thesis] (1988), Ontario, Canada: University of Waterloo, Ontario, Canada
[4] Biegler-König, F. W., Construction of band matrices from spectral data, Linear Algebra and Its Applications, 40, 79-87 (1981) · Zbl 0468.15006
[5] de Boor, C.; Golub, G. H., The numerically stable reconstruction of a Jacobi matrix from spectral data, Linear Algebra and Its Applications, 21, 3, 245-260 (1978) · Zbl 0388.15010
[6] Gladwell, G. M. L.; Willms, N. B., A discrete Gel’fand-Levitan method for band-matrix inverse eigenvalue problems, Inverse Problems, 5, 2, 165-179 (1989) · Zbl 0673.65019
[7] Boley, D.; Golub, G. H.; Watson, G. A., Inverse eigenvalue problems for band matrices, Numerical Analysis (Proc. 7th Biennial Conf., Univ. Dundee, Dundee, 1977). Numerical Analysis (Proc. 7th Biennial Conf., Univ. Dundee, Dundee, 1977), Lecture Notes in Math., 630, 23-31 (1978), Berlin, Germany: Springer, Berlin, Germany · Zbl 0367.65023
[8] Hald, O. H., Inverse eigenvalue problems for Jacobi matrices, Linear Algebra and Its Applications, 14, 1, 63-85 (1976) · Zbl 0328.15007
[9] Hochstadt, H., On the construction of a Jacobi matrix from spectral data, Linear Algebra and Its Applications, 8, 435-446 (1974) · Zbl 0288.15029
[10] Golub, G. H.; Van Loan, C. F., Matrix Computations. Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences, xiv+756 (2013), Baltimore, Md, USA: Johns Hopkins University Press, Baltimore, Md, USA · Zbl 1268.65037
[11] Hwang, S.-G., Cauchy’s interlace theorem for eigenvalues of Hermitian matrices, The American Mathematical Monthly, 111, 2, 157-159 (2004) · Zbl 1050.15008
[12] Fisk, S., A very short proof of Cauchy’s interlace theorem for eigenvalues of Hermitian matrices, The American Mathematical Monthly, 112, 2, 118 (2005)
[13] De Serre Rothney, A., Eigenvalues of a special tridiagonal matrix
[14] Clement, P. A., A class of triple-diagonal matrices for test purposes, SIAM Review, 1, 50-52 (1959) · Zbl 0117.14202
[15] Edelman, A.; Kostlan, E.; In, ., The road from Kac’s matrix to Kac’s random polynomials, Proceedings of the SIAM Applied Linear Algebra Conference · Zbl 0817.15014
[16] Muir, T., A Treatise on the Theory of Determinants, vii+766 (1960), New York, NY, USA: Dover, New York, NY, USA
[17] Taussky, O.; Todd, J., Another look at a matrix of Mark Kac, Linear Algebra and Its Applications, 150, 341-360 (1991) · Zbl 0727.15010
[18] Nikiforov, A. F.; Suslov, S. K.; Uvarov, V. B., Classical Orthogonal Polynomials of a Discrete Variable. Classical Orthogonal Polynomials of a Discrete Variable, Springer Series in Computational Physics, xvi+374 (1991), Berlin, Germany: Springer, Berlin, Germany · Zbl 0743.33001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.