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Funktsional. Anal. i Prilozhen., 2004, Volume 38, Issue 1, Pages 85–88 (Mi faa100)  

This article is cited in 25 scientific papers (total in 25 papers)

Brief communications

On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank

S. V. Savchenko

L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

Abstract: We show that the $r$ largest Jordan blocks disappear and all other blocks remain the same in the part of the Jordan form corresponding to a given eigenvalue $\lambda$ under a generic rank $r$ perturbation. Moreover, a necessary and sufficient condition on the entries of a perturbation under which the spectral properties of $\lambda$ change in this manner is obtained with the use of the resolvent technique for the case in which the geometric multiplicity of $\lambda$ is greater than or equal to $r$. A Jordan basis in the corresponding root space is constructed from the Jordan chains of the original matrix. A complete description of how the spectrum changes in a small neighborhood of the point $z=\lambda$ is given for the case of a small parameter multiplying the perturbation.

Keywords: generic rank $r$ perturbation, scalar resolvent matrix, root space, Jordan block, Jordan basis, Binet–Cauchy formula, Laurent series

DOI: https://doi.org/10.4213/faa100

Full text: PDF file (151 kB)
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English version:
Functional Analysis and Its Applications, 2004, 38:1, 69–71

Bibliographic databases:

UDC: 512.643+517.983
Received: 03.10.2002

Citation: S. V. Savchenko, “On the Change in the Spectral Properties of a Matrix under Perturbations of Sufficiently Low Rank”, Funktsional. Anal. i Prilozhen., 38:1 (2004), 85–88; Funct. Anal. Appl., 38:1 (2004), 69–71

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. S. V. Savchenko, “Laurent expansion for the determinant of the matrix of scalar resolvents”, Sb. Math., 196:5 (2005), 743–764  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. De Terán F., Dopico F.M., “Low rank perturbation of Kronecker structures without full rank”, SIAM J. Matrix Anal. Appl., 29:2 (2006), 496–529  crossref  mathscinet  isi  scopus
    3. De Terán F., Dopico F.M., Moro J., “Low rank perturbation of Weierstrass structure”, SIAM J. Matrix Anal. Appl., 30:2 (2008), 538–547  crossref  mathscinet  zmath  isi  scopus
    4. Glebsky L., Rivera L.M., “On low rank perturbations of complex matrices and some discrete metric spaces”, Electron. J. Linear Algebra, 18 (2009), 302–316  crossref  mathscinet  zmath  isi
    5. Mehl Ch., Mehrmann V., Ran A.C.M., Rodman L., “Eigenvalue perturbation theory of classes of structured matrices under generic structured rank one perturbations”, Linear Algebra Appl, 435:3 (2011), 687–716  crossref  mathscinet  zmath  isi  scopus
    6. Ran A.C.M., Wojtylak M., “Eigenvalues of Rank One Perturbations of Unstructured Matrices”, Linear Alg. Appl., 437:2 (2012), 589–600  crossref  mathscinet  zmath  isi  elib  scopus
    7. Mehl Ch., Mehrmann V., Ran A.C.M., Rodman L., “Perturbation Theory of Selfadjoint Matrices and Sign Characteristics Under Generic Structured Rank One Perturbations”, Linear Alg. Appl., 436:10, SI (2012), 4027–4042  crossref  mathscinet  zmath  isi  scopus
    8. S. M. Tashpulatov, “Spectrum of Two-Magnon non-Heisenberg Ferromagnetic Model of Arbitrary Spin with Impurity”, Zhurn. matem. fiz., anal., geom., 9:2 (2013), 239–265  mathnet  mathscinet
    9. Mehl Ch., Mehrmann V., Ran A.C.M., Rodman L., “Jordan Forms of Real and Complex Matrices Under Rank One Perturbations”, Oper. Matrices, 7:2 (2013), 381–398  crossref  mathscinet  zmath  isi  scopus
    10. Han L., Xu J., “Proof of Stenger's Conjecture on Matrix I(-1) of Sinc Methods”, J. Comput. Appl. Math., 255 (2014), 805–811  crossref  mathscinet  zmath  isi  elib  scopus
    11. Batzke L., “Generic Rank-One Perturbations of Structured Regular Matrix Pencils”, Linear Alg. Appl., 458 (2014), 638–670  crossref  mathscinet  zmath  isi  scopus
    12. Bierkens J., Ran A., “A Singular M-Matrix Perturbed By a Nonnegative Rank One Matrix Has Positive Principal Minors; Is It D-Stable?”, Linear Alg. Appl., 457 (2014), 191–208  crossref  mathscinet  zmath  isi  scopus
    13. Mehl Ch., Mehrmann V., Ran A.C.M., Rodman L., “Eigenvalue Perturbation Theory of Symplectic, Orthogonal, and Unitary Matrices Under Generic Structured Rank One Perturbations”, Bit, 54:1 (2014), 219–255  crossref  mathscinet  zmath  isi  scopus
    14. Mehl Ch., Mehrmann V., Wojtylak M., “on the Distance To Singularity Via Low Rank Perturbations”, Oper. Matrices, 9:4 (2015), 733–772  crossref  mathscinet  zmath  isi  scopus
    15. Batzke L., “Sign Characteristics of Regular Hermitian Matrix Pencils Under Generic Rank-1 Perturbations and a Certain Class of Generic Rank-2 Perturbations”, Electron. J. Linear Algebra, 30 (2015), 760–794  crossref  mathscinet  zmath  isi  scopus
    16. Behrndt J., Leben L., Martinez Peria F., Trunk C., “the Effect of Finite Rank Perturbations on Jordan Chains of Linear Operators”, Linear Alg. Appl., 479 (2015), 118–130  crossref  mathscinet  zmath  isi  elib  scopus
    17. Mehl Ch., Mehrmann V., Ran A.C.M., Rodman L., “Eigenvalue Perturbation Theory of Structured Real Matrices and Their Sign Characteristics Under Generic Structured Rank-One Perturbations”, Linear Multilinear Algebra, 64:3 (2016), 527–556  crossref  mathscinet  zmath  isi  scopus
    18. Kohaupt L., “Further spectral properties of the matrix
      $$C^{-1} B$$
      C - 1 B with positive definite C and Hermitian B applied to wider classes of matrices C and B”, J. Appl. Math. Comput., 52:1-2 (2016), 215–243  crossref  mathscinet  zmath  isi  elib  scopus
    19. Batzke L., “Generic rank-two perturbations of structured regular matrix pencils”, Oper. Matrices, 10:1 (2016), 83–112  crossref  mathscinet  zmath  isi  scopus
    20. Kohaupt L., “Spectral properties of the matrix
      $$C^{-1} B$$
      C - 1 B with positive definite
      $$C$$
      C and Hermitian
      $$B$$
      B as well as applications”, J. Appl. Math. Comput., 50:1-2 (2016), 389–416  crossref  mathscinet  zmath  isi  elib  scopus
    21. Sosa F., Moro J., “First Order Asymptotic Expansions for Eigenvalues of Multiplicatively Perturbed Matrices”, SIAM J. Matrix Anal. Appl., 37:4 (2016), 1478–1504  crossref  mathscinet  zmath  isi  scopus
    22. De Teran F., Dopico F.M., “Generic Change of the Partial Multiplicities of Regular Matrix Pencils under Low-Rank Perturbations”, SIAM J. Matrix Anal. Appl., 37:3 (2016), 823–835  crossref  mathscinet  zmath  isi  scopus
    23. Gernandt H., Trunk C., “Eigenvalue Placement for Regular Matrix Pencils with Rank One Perturbations”, SIAM J. Matrix Anal. Appl., 38:1 (2017), 134–154  crossref  mathscinet  zmath  isi  scopus
    24. Kula A., Wojtylak M., Wysoczanski J., “Rank two perturbations of matrices and operators and operator model for t-transformation of probability measures”, J. Funct. Anal., 272:3 (2017), 1147–1181  crossref  mathscinet  zmath  isi  scopus
    25. Mehl Ch., Ran A.C.M., “Low Rank Perturbations of Quaternion Matrices”, Electron. J. Linear Algebra, 32 (2017), 514–530  crossref  mathscinet  zmath  isi  scopus
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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