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 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.**5**(1963), 73–87. MR**0159412****[3]**E. E. Osborne,*On pre-conditioning of matrices*, J. Assoc. Comput. Mach.**7**(1960), 338–345. MR**0143333****[4]**J. Stoer and C. Witzgall,*Transformations by diagonal matrices in a normed space*, Numer. Math.**4**(1962), 158–171. MR**0150151****[5]**J. H. Wilkinson,*The algebraic eigenvalue problem*, Clarendon Press, Oxford, 1965. MR**0184422**

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

Article copyright:
© Copyright 1969
American Mathematical Society