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Extremal properties of balanced tri-diagonal matrices

Author: Peter A. Businger
Journal: Math. Comp. 23 (1969), 193-195
MSC: Primary 65.35
MathSciNet review: 0238476
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Abstract: If $ A$ is a square matrix with distinct eigenvalues and $ D$ a nonsingular matrix, then the angles between row- and column-eigenvectors of $ {D^{ - 1}}AD$ differ from the corresponding quantities of $ A$. Perturbation analysis of the eigenvalue problem motivates the minimization of functions of these angles over the set of diagonal similarity transforms; two such functions which are of particular interest are the spectral and the Euclidean condition numbers of the eigenvector matrix $ X$ of $ {D^{ - 1}}AD$. It is shown that for a tri-diagonal real matrix $ A$ both these condition numbers are minimized when $ D$ is chosen such that the magnitudes of corresponding sub- and super-diagonal elements are equal.

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

  • [1] F. L. Bauer, ``Some aspects of scaling invariance,'' Colloq. Internat. C.N.R.S., No. 165, pp. 37-47.
  • [2] F. L. Bauer, ``Optimally scaled matrices,'' Numer. Math., v. 5, 1963, pp. 73-87. MR 28 #2629. MR 0159412 (28:2629)
  • [3] E. E. Osborne, ``On pre-conditioning of matrices,'' J. Assoc. Comput. Mach., v. 7, 1960, pp. 338-345. MR 26 #892. MR 0143333 (26:892)
  • [4] J. Stoer & Ch. Witzgall, ``Transformations by diagonal matrices in a normed space,'' Numer. Math., v. 4, 1962, pp. 158-171. MR 27 #154. MR 0150151 (27:154)
  • [5] J. H. Wilkinson, The Algebraic Eigenvalue Problem, Clarendon Press, Oxford, 1965. MR 32 #1894. MR 0184422 (32:1894)

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Article copyright: © Copyright 1969 American Mathematical Society

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