On the Scalar Measure of Non-Normality of Matrices - Dimension vs. Structure

The aim of this paper is to analyze the relative importance of the dimension and the structure of the square matrices in the quantification of their non-normality. We envisage non-normal matrices which come from numerical analysis of o. d. e. and p. d. e. as well as from various iterative processes where the parameter dimension varies. The main result consists in an upper bound for the departure from normality. This bound is a product of two factors, is based directly on the entries of the matrices, and is elementary computable. The first factor depends exclusively on the dimension of matrices and the second, called the aspect factor, is intimately related to the structure of the matrices. In some special situations, the aspect factor is independent of the dimension, and shows that, in these cases, only the dimension is responsible for the departure from normality. 21 22 Călin-Ioan Gheorghiu An upper bound for the field of values is also obtained. Some numerical experiments are carried out. They underline the idea that the aspect factor and pseudospectrum are complementary aspects of the non-normality. 2000 Mathematical Subject Classification:65F35, 65F50, 15A60, 15A12.