Extremal properties of balanced tri-diagonal matrices

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.