Abstract of the paper sent to SIAM J. Sci. Comput.
"A New Scalar Measure of Matrices' Non-Normality. Application to the Characterization of Some Spectral Methods"
C. I. Gheorghiu
key words: non-normal matrices; scalar measure; Chebyshev-type methods; pseudospectrum; Helmholtz equation; 1D biharmonic equation;
AMS subject classification: 65F15; 65F35; 65L10; 65L60;

The aim of the paper is twofold. First, we introduce a new euclidean relative departure from normality of a square matrix and find a practical upper bound for that. This bound is a product of two factors. The first one depends exclusively on the dimension n of the matrix and has the order O(n^3/4). The latter, called the non-normality ratio, depends mainly on the structure of the matrix and ranges from 0, just in case of normal matrices, to 2^1/4. The result is sharp in the sense that the matrices corresponding to the upper bound are known. Moreover, this ratio is itself a scalar measure of non-normality which will be exclusively exploited in our numerical experiments. The existence of the above upper bound makes the difference between the non-normality ratio and scalar measures existent in literature. An upper bound for a relative distance of a matrix to the set of normal matrices is also provided. Second, we exemplify the capabilities of the introduced ratio by analyzing three spectral methods of Chebyshev type with respect to the normality of associated system matrices. The matrices are attached to differential operators involved in the unidimensional second order and fourth order two-point boundary value problems. In a sequence of tables the non-normality ratios are displayed for typical choices of parameters entering these problems. They underline the fact that our non-normality ratio considered as a measure furnishes a hierarchy for a specific class of numerical methods and moreover, thoroughly observe the non-normality even if the pseudospectra fail to take notice of that.