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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

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.


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DOI: http://dx.doi.org/10.1090/S0025-5718-1969-0238476-6
PII: S 0025-5718(1969)0238476-6
Article copyright: © Copyright 1969 American Mathematical Society