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Mathematics of Computation

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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.

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Article copyright: © Copyright 1969 American Mathematical Society