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An application of the separation theorem for hermitian matrices


Author: T. L. Markham
Journal: Proc. Amer. Math. Soc. 47 (1975), 61-64
MSC: Primary 15A18
DOI: https://doi.org/10.1090/S0002-9939-1975-0364290-1
MathSciNet review: 0364290
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Abstract: Suppose $ H$ is an $ n \times n$ hermitian matrix over the complex field partitioned as $ H = \left(\begin{smallmatrix}A&B\\ B*&C\end{smallmatrix}\right)$, where $ C$ is invertible. Using the separation theorem on eigenvalues of hermitian matrices, bounds are obtained for the eigenvalues of $ (H/C) = A - B{C^{ - 1}}{B^ \ast }$ in terms of the eigenvalues of $ H$ and $ C$.


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

  • [1] D. E. Crabtree and E. V. Haynsworth, An identity for the Schur complement of a matrix, Proc. Amer. Math. Soc. 22 (1969), 364-366. MR 41 #234. MR 0255573 (41:234)
  • [2] E. V. Haynsworth, Determination of the inertia of a partitioned Hermitian matrix, Linear Algebra and Appl. 1 (1968), 73-81. MR 36 #6440. MR 0223392 (36:6440)
  • [3] Alston A. Householder, The theory of matrices in numerical analysis, Blaisdell, New York, 1964, p. 76. MR 30 #5475. MR 0175290 (30:5475)
  • [4] M. Marcus and H. Minc, A survey of matrix theory and matrix inequalities, Allyn and Bacon, Boston, Mass., 1964. MR 29 #112. MR 0162808 (29:112)
  • [5] L. J. Watford, Jr., The Schur complement of a generalized $ M$-matrix, Linear Algebra and Appl. 5 (1972), 247-255. MR 46 #9075. MR 0309972 (46:9075)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1975-0364290-1
Keywords: Separation theorem, hermitian matrices, Schur complement, compound matrix, bounds for eigenvalues
Article copyright: © Copyright 1975 American Mathematical Society

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