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Eigenvalues of graded matrices and the condition numbers of a multiple eigenvalue

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Summary

This paper concerns two closely related topics: the behavior of the eigenvalues of graded matrices and the perturbation of a nondefective multiple eigenvalue. We will show that the eigenvalues of a graded matrix tend to share the graded structure of the matrix and give precise conditions insuring that this tendency is realized. These results are then applied to show that the secants of the canonical angles between the left and right invariant of a multiple eigenvalue tend to characterize its behavior when its matrix is slightly perturbed.

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References

  1. Barlow, J., Demmel, J.: Computing accurate eigensystems of scaled diagonally dominant matrices. Technical report 421, Computer Science Department, Courant Institute (1988). SIAM J. Numer. Anal. (to appear)

  2. J. Demmel, J.: The smallest perturbation of a submatrix which lowers the rank and constrained total least squares problems. SIAM J. Numer. Anal.24, 199–206 (1987)

    Google Scholar 

  3. Demmel, J., Veselic, K.: Jacobi's method is more accurate than QR. Technical report 468, Computer Science Department, New York University, 1989

  4. Golub, G., Hoffman, A., Stewart, G.W.: A Generalization of the Echart-Young Matrix Approximation Theorem. Linear Algebra Appl.88/89, 317–327 (1987)

    Google Scholar 

  5. Golub, G.H., Van Loan, C.F.: Matrix computations, 2nd ed. Baltimore, Maryland: Johns Hopkins University Press 1989

    Google Scholar 

  6. Stewart, G.W.: Error and perturbation bounds for subspaces associated with certain eigenvalue problems. SIAM Review15, 727–764 (1973)

    Google Scholar 

  7. Stewart, G.W.: On the Asymptotic Behavior of Scaled Singular Value and QR Decompositions. Math. Comput.43, 483–489 (1984)

    Google Scholar 

  8. Stewart, G.W., Sun, J.-G.: Matrix perturbation theory. Boston: Academic Press 1990

    Google Scholar 

  9. Sun, J.-G.: A note on local behavior of multiple eigenvalues. SIAM J. Matrix Anal. Appl.10, 533–541 (1989)

    Google Scholar 

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Authors and Affiliations

  1. Department of Computer Science, University of Maryland, 20742, College Park, MD, USA

    G. W. Stewart

  2. Institute for Advanced Computer Studies, University of Maryland, 20742, College Park, MD, USA

    G. W. Stewart & G. Zhang

Authors
  1. G. W. Stewart
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  2. G. Zhang
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Additional information

This work was supported in part by the Air Force Office of Sponsored Research under Contract AFOSR-87-0188

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Stewart, G.W., Zhang, G. Eigenvalues of graded matrices and the condition numbers of a multiple eigenvalue. Numer. Math. 58, 703–712 (1990). https://doi.org/10.1007/BF01385650

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  • DOI: https://doi.org/10.1007/BF01385650

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