On the Stability of Uniformly Asymptotically Diagonal Systems

In a number of papers ([1], [2]), Delves and Mead have derived some useful (though limited) rate of convergence results which can be applied to variational approximations for the solution of linear positive definite operator equations when the coordinate system is uniformly asymptotically diagonal. Independently, Mikhlin [5] has examined the stability of such variational approximations in the case of positive definite operators and concluded that the use of strongly minimal coordinate systems is a necessary and sufficient condition for their stability. Since, in general, the Delves and Mead results will only be applicable to actual variational approximations when their uniformly asymptotically diagonal system is at least strongly minimal, we examine the properties of uniformly asymptotically diagonal systems in terms of the minimal classification of Mikhlin. We show that (a) a normalized uniformly asymptotically diagonal system is either nonstrongly minimal or almost orthonormal; (b) the largest eigenvalue of a normalized uniformly asymptotically diagonal system is bounded above, independently of the size of the system; (c) the special property of normalized uniformly asymptotically diagonal systems mentioned in (b) is often insufficient to prevent their yielding unstable results when these systems are not strongly minimal.