Abstract
For standard eigenvalue problems, closed-form expressions for the condition numbers of a multiple eigenvalue are known. In particular, they are uniformly 1 in the Hermitian case and generally take different values in the non-Hermitian case. We consider the generalized eigenvalue problem and identify the condition numbers. Our main result is that a multiple eigenvalue generally has multiple condition numbers, even in the Hermitian definite case. The condition numbers are characterized in terms of the singular values of the outer product of the corresponding left and right eigenvectors.
Similar content being viewed by others
References
Barlow J., Demmel J.: Computing accurate eigensystems of scaled diagonally dominant matrices. SIAM J. Numer. Anal. 27(3), 762–791 (1990)
De Teran F., Dopico F.M., Moro J.: First order spectral perturbation theory of square singular matrix pencils. Linear Algebra Appl. 429(2–3), 548–576 (2008)
Demmel J., Veselic K.: Jacobi’s method is more accurate than QR. SIAM J. Matrix Anal. Appl. 13(4), 1204–1245 (1992)
Golub G.H., Van Loan C.F.: Matrix Computations, 3rd edn. The Johns Hopkins University Press, Baltimore (1996)
Higham D.J., Higham N.J.: Structured backward error and condition of generalized eigenvalue problems. SIAM J. Matrix Anal. Appl. 20(2), 493–512 (1998)
Kressner D., JosePelaez M., Moro J.: Structured Hölder condition numbers for multiple eigenvalues. SIAM J. Matrix Anal. Appl. 31(1), 175–201 (2009)
Li R.C., Nakatsukasa Y., Truhar N., Xu S.: Perturbation of partitioned Hermitian definite generalized eigenvalue problems. SIAM J. Matrix Anal. Appl. 32(2), 642–663 (2011)
Moro J., Burke J.V., Overton M.L.: On the Lidskii–Vishik–Lyusternik perturbation theory for eigenvalues of matrices with arbitrary Jordan structure. SIAM J. Matrix Anal. Appl. 18(4), 793–817 (1997)
Nakatsukasa Y.: Perturbation behavior of a multiple eigenvalue in generalized Hermitian eigenvalue problems. BIT 50(1), 109–121 (2010)
Stewart G.W.: Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Rev. 15(4), 727–764 (1973)
Stewart, G.W.: Matrix Algorithms. Basic Decompositions, vol. I. SIAM, Philadelphia (1998)
Stewart, G.W.: Matrix Algorithms. Eigensystems, vol. II, SIAM, Philadelphia (2001)
Stewart G.W., Sun J.G.: Matrix Perturbation Theory. Academic Press, New York (1990)
Stewart G.W., Zhang G.: Eigenvalues of graded matrices and the condition numbers of a multiple eigenvalue. Numer. Math. 58(7), 703–712 (1991)
Sun J.G.: On condition numbers of a nondefective multiple eigenvalue. Numer. Math. 61(2), 265–275 (1992)
Sun J.G.: On worst-case condition numbers of a nondefective multiple eigenvalue. Numer. Math. 69(3), 373–382 (1995)
Veselic K., Slapnicar I.: Floating-point perturbations of Hermitian matrices. Linear Algebra Appl. 195, 81–116 (1993)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nakatsukasa, Y. On the condition numbers of a multiple eigenvalue of a generalized eigenvalue problem. Numer. Math. 121, 531–544 (2012). https://doi.org/10.1007/s00211-011-0440-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-011-0440-x