Principal Direction Alignment, a Geometric Framework for the Computer Solution to the mu-Problem

摘要

Nonsimilarity scaling results for element by element bounded perturbations, particularly the Major Principal Direction Alignment (MPDA) property, are reviewed. The block diagonal bounded perturbation problem formulation of Doyle (1982) for the case of 1x1 blocks is summarized. It is proved that a computable solution for mu is always guaranteed under MPDA, for this class of perturbation. A simple extension which allows the treatment of 1x1 blocks with different upper bounds is presented. General block structured perturbations are examined, and using the MPDA principle, a solution to the mu problem is presented. The case of repeated singular values is considered. The generality of the principal direction alignment property is established.