Extremal properties of balanced tri-diagonal matrices

Author:
Peter A. Businger

Journal:
Math. Comp. **23** (1969), 193-195

MSC:
Primary 65.35

DOI:
https://doi.org/10.1090/S0025-5718-1969-0238476-6

MathSciNet review:
0238476

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Abstract: If is a square matrix with distinct eigenvalues and a nonsingular matrix, then the angles between row- and column-eigenvectors of differ from the corresponding quantities of . 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 of . It is shown that for a tri-diagonal real matrix both these condition numbers are minimized when is chosen such that the magnitudes of corresponding sub- and super-diagonal elements are equal.

**[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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DOI:
https://doi.org/10.1090/S0025-5718-1969-0238476-6

Article copyright:
© Copyright 1969
American Mathematical Society