Extremal properties of balanced tri-diagonal matrices

Abstract

If A A is a square matrix with distinct eigenvalues and D D a nonsingular matrix, then the angles between row- and column-eigenvectors of D − 1 A D {D^{ - 1}}AD differ from the corresponding quantities of A 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 X of D − 1 A D {D^{ - 1}}AD . It is shown that for a tri-diagonal real matrix A A both these condition numbers are minimized when D D is chosen such that the magnitudes of corresponding sub- and super-diagonal elements are equal.