Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



On perturbations of matrix pencils
with real spectra. II

Authors: Rajendra Bhatia and Ren-Cang Li
Journal: Math. Comp. 65 (1996), 637-645
MSC (1991): Primary 15A22, 15A42, 65F15
MathSciNet review: 1333304
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A well-known result on spectral variation of a Hermitian matrix due to Mirsky is the following: Let $A$ and $% \widetilde A$ be two $n\times n$ Hermitian matrices, and let $% \lambda_1,\ldots,\lambda_{n}$ and $% \widetilde\lambda_1,\ldots,\widetilde\lambda_{n}$ be their eigenvalues arranged in ascending order. Then $% \left|\kern-1.5truept\left|\kern-1.5truept\left| \operatorname{diag}\, (\lambda_1-% \widetilde\lambda_1,\ldots,\lambda_n-% \widetilde\lambda_n)% \right|\kern-1.5truept\right|\kern-1.5truept\right| \le% \left|\kern-1.5truept\left|\kern-1.5truept\left| A-\widetilde A% \right|\kern-1.5truept\right|\kern-1.5truept\right|$ for any unitarily invariant norm $% \hbox{$|\kern-1.5truept|\kern-1.5truept|\cdot |\kern-1.5truept|\kern-1.5truept|$}$. In this paper, we generalize this to the perturbation theory for diagonalizable matrix pencils with real spectra. The much studied case of definite pencils is included in this.

References [Enhancements On Off] (What's this?)

  • 1. R. Bhatia, Perturbation bounds for matrix eigenvalues, Longman, Essex and Wiley, New York, 1987. MR 88k:15020
  • 2. R. Bhatia and Ch. Davis, A bound for the spectral variation of a unitary operator, Linear Multilinear Algebra 15(1984), 71--76. MR 85b:15020
  • 3. R. Bhatia, Ch. Davis and F. Kittaneh, Some inequalities for commutators and an application to spectral variation, Aequationes Mathematicae, 41(1991), 70--78. MR 92j:47023
  • 4. R. Bhatia, Ch. Davis and A. McIntosh, Perturbation of spectral subspaces and solution of linear operator equations, Linear Algebra Appl., 52/53(1983), 45--67. MR 85a:47020
  • 5. C. R. Crawford, A stable generalized eigenvalue problem, SIAM J. Numer. Anal., 8(1976), 854--860. MR 55:4628
  • 6. Ch. Davis, and W. M. Kahan, The rotation of eigenvectors by a perturbation. III, SIAM J. Numer. Anal., 7(1970), 1--46. MR 41:9044
  • 7. L. Elsner and J.-G. Sun, Perturbation theorems for the generalized eigenvalue problem, Linear Algebra Appl., 48(1982), 341--357. MR 84f:15012
  • 8. A. J. Hoffman and H. W. Wielandt, The variation of the spectrum of a normal matrix, Duke Math. J., 20(1953), 37--39. MR 14:611b
  • 9. R.-C. Li, On perturbations of matrix pencils with real spectra, Math. Comp., 62(1994), 231--265. MR 94f:15011
  • 10. ------, Norms of certain matrices with applications to variations of the spectra of matrices and matrix pencils, Linear Algebra Appl., 182(1993), 199--234. MR 94c:15040
  • 11. ------, A perturbation bound for definite pencils, Linear Algebra Appl., 179(1993), 191--202. MR 94h:15007
  • 12. V. B. Lidskii, The proper values of the sum and product of symmetric matrices, Doklady Akademii Nauk SSSR, 75(1950), 769--772. In Russian. Translation by C. Benster available from the National Translation Center of the Library of Congress. MR 14:528i
  • 13. K. Loewner, Über monotone Matrixfunktionen, Math. Z., 38(1934), 177--216.
  • 14. L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford, 11(1960), 50--59. MR 22:5639
  • 15. G. W. Stewart, Perturbation bounds for the definite generalized eigenvalue problem, Linear Algebra Appl., 23(1979), 69--85. MR 80c:15007
  • 16. G. W. Stewart and J.-G. Sun, Matrix Perturbation Theory, Academic Press, New York, 1990. MR 92a:65017
  • 17. J.-G. Sun, A note on Stewart's theorem for definite matrix pairs, Linear Algebra Appl., 48(1982), 331--339. MR 84f:15013
  • 18. ------, The perturbation bounds for eigenspaces of a definite matrix-pair, Numer. Math., 41(1983), 321--343. MR 85c:65045
  • 19. ------, The perturbation bounds of generalized eigenvalues of a class of matrix-pairs, Math. Numer. Sinica, 4(1982), 23--29 (Chinese). MR 85h:15021
  • 20. ------, On the perturbation of the eigenvalues of a normal matrix, Math. Numer. Sinica, 6(1984), 334-336 (Chinese). MR 86d:15010
  • 21. H. Weyl, Das asymptotische Verteilungsgesetz der Eigenwerte linearer partieller Differentialgleichungen (mit einer Anwendung auf die Theorie der Hohlraumstrahlung), Mathematische Annalen, 71(1912), 441--479.
  • 22. H. Wielandt, An extremum property of sums of eigenvalues, Proc. Amer. Math. Soc., 6(1955), 106--110. MR 16:785a

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 15A22, 15A42, 65F15

Retrieve articles in all journals with MSC (1991): 15A22, 15A42, 65F15

Additional Information

Rajendra Bhatia
Affiliation: Indian Statistical Institute, 7, S. J. S. Sansanwal Marg, New Delhi – 110016, India

Ren-Cang Li
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720

Keywords: Diagonalizable matrix pencil, definite pencil, real spectrum, unitarily invariant norm, perturbation bound
Received by editor(s): April 13, 1994
Received by editor(s) in revised form: August 4, 1994
Additional Notes: The first author thanks NSERC Canada for financial support.
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society