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The distance between the eigenvalues of Hermitian matrices

Author: Rajendra Bhatia
Journal: Proc. Amer. Math. Soc. 96 (1986), 41-42
MSC: Primary 15A42; Secondary 15A60
MathSciNet review: 813806
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Abstract: It is shown that the minmax principle of Ky Fan leads to a quick simple derivation of a recent inequality of V. S. Sunder giving a lower bound for the spectral distance between two Hermitian matrices. This brings out a striking parallel between this result and an earlier known upper bound for the spectral distance due to L. Mirsky.

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

  • [1] R. Bhatia, Analysis of spectral variation and some inequalities, Trans. Amer. Math. Soc. 272 (1982), 323-331. MR 656492 (83k:15015)
  • [2] Ky Fan, On a theorem of Weyl concerning eigenvalues of linear transformations. I, Proc. Nat. Acad. Sci. U.S.A. 35 (1949), 652-655. MR 0034519 (11:600e)
  • [3] A. W. Marshall and I. Olkin, Inequalities: Theory of majorisation and its applications, Academic Press, New York, 1979. MR 552278 (81b:00002)
  • [4] L. Mirsky, Symmetric gauge functions and unitarily invariant norms, Quart. J. Math. Oxford (2) 11 (1960), 50-59. MR 0114821 (22:5639)
  • [5] V. S. Sunder, On permutations, convex hulls and normal operators, Linear Algebra and Appl. 48 (1982), 403-411. MR 683234 (85b:15032)

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Keywords: Hermitian matrices, eigenvalues, minmax principle, majorisation
Article copyright: © Copyright 1986 American Mathematical Society

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