Repeated eigenvalues of a matrix with elements polynomially dependent on a parameter

E. A. Kalinina

Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, no. 2 (2016), pp. 26-32

Résumé

Earlier matrices with multiple eigenvalues were considered only for theoretical purposes. However, now such non-generic matrices are also of practical interest because they appear in different problems of quantum mechanics, nuclear physics, optics and dynamic of mechanical systems. A square matrix with elements that are linearly dependent on a parameter is considered in this paper. A method to find the values of the parameter such that the matrix has a repeated eigenvalue is considered. We find the polynomial whose roots are these values of the parameter using only finite number of algebraic operations on the matrix elements. The method can be generalized to matrices with elements algebraically dependent on a parameter. A numerical example of how to find the required parameter values is considered. Refs 16.

Détail

Keywords: the Kronecker product, the Leverrier method, the Newton sums.

Bibliographie

[1] Kalinina E. A., “Common eigenvalues of two matrices”, Far Eastern Mathematical Journal, 13:1 (2013), 52–60 (In Russian) | MR | Zbl

[2] Mailybaev A. A., “Computation of multiple eigenvalues and generalized eigenvectors for matrices dependent on parameters”, Numer. Linear Algebra Appl., 13 (2006), 419–436 | DOI | MR | Zbl

[3] Schucan T. H., Weidenmüller H. A., “Perturbation theory for the effective interaction in nuclei”, Ann. Physics, 76 (1973), 483–501 | DOI | MR

[4] Karow M., “Eigenvalue condition numbers and a formula of Burke, Lewis and Overton”, Electron. J. Linear Algebra, 15 (2006), 143–153 | DOI | MR | Zbl

[5] Kressner D., Peláez M. J., Moro J., “Structured Hölder condition numbers for multiple eigenvalues”, SIAM J. Matrix Anal. Appl., 31:1 (2009), 175–201 | DOI | MR | Zbl

[6] Burke J. V., Lewis A. S., Overton M., “Optimization and pseudospectra, with applications to robust stability”, SIAM J. Matrix Anal. Appl., 25:1 (2003), 80–104 | DOI | MR | Zbl

[7] Jarlebring E., Kvaal S., Michiels W., “Computing all pairs $(\lambda;\mu)$ such that $\lambda$ is a double eigenvalue of $A+\mu B$”, SIAM J. Matrix Anal. Appl., 32 (2011), 902–927 | DOI | MR | Zbl

[8] Muhič A., Plestenjak B., “A method for computing all values $\lambda$ such that $A+\lambda B$ has a multiple eigenvalue”, Linear Algebra Appl., 440 (2014), 345–359 | DOI | MR | Zbl

[9] Faddeev D. K., Faddeeva V. N., Numerical methods of linear algebra Moscow, GIFML Publ., 1960, 656 pp. (In Russian)

[10] Littlewood D. E., The theory of group characters and matrix representations of groups, Oxford University Press, Oxford, 1950, 310 pp. | MR | Zbl

[11] Horn R. A., Johnson Ch. R., Topics in matrix analysis, Cambridge University Press, New York, 1991, 607 pp. | MR | Zbl

[12] MacDuffee C. C., The Theory of Matrices, Chelsea Publ. Company, New York, 1956, 110 pp.

[13] Faddeev D. K., Lections in algebra, Nauka Publ., M., 1984, 416 pp. (In Russian)

[14] Gantmacher F. R., Theory of matrices, v. 2, AMS Chelsea Publ. Company, Providence, RI, 1960, 276 pp. | MR

[15] Strassen V., “Gaussian Elimination is not Optimal”, Numerische Mathematik, 13 (1969), 354–356 | DOI | MR | Zbl

[16] Wilkinson J. H., The algebraic eigenvalue problem, Clarendon Press, Oxford, 1965, 662 pp. | MR | Zbl