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 ODEs and PDEs, 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. 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.


Călin-Ioan Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis


matrix; complex entry; normality measure; field of values; upper bound; aspect factor;


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Cite this paper as

C.I. Gheorghiu, On the scalar measure of non-normality of matrices – dimension vs. structure, General Mathematics, 11 (2003) nos. 1–2, 21–32.


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“Lucian Blaga”  University of Sibiu
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[1] C. Canuto, M.Y. Hussaini, A. Quarteroni, T.A. Zang, Spectral Methods in Fluid Dynamics, Springer Verlag, 1988 .
[2] F. Chaitin-Chatelin, V. Fraisse, Lectures on Finite Precision Computation, SIAM Philadelphia, 1996 .
[3] D. Gotlieb, S.A. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAM Philadelphia, 1997
[4] P. Henrici, Bounds for iterates, inverses, spectral variation and fields of values of non-normal matrices, Numer. Math. 4, 24-40 (1962) .
[5] S.L. Lee, A Practical Upper Bound for Departure from Normality, SIAM J. Matrix Anal. Appl., Vol. 16, No. 2, pp462-468, April 1995
[6] P.J. Smith, D.S. Henningson, Stability and Transition in Shear Flows, Springer Verlag, 2001
[7] B.J. Stone, Best possible ratios of certain matrix norms, Numer. Math. 4, 114-116(1962)
[8] L.N. Trefethen, Pseudospectra of linear operators, SIAM Review, 39:383-406, (1997)
[9] L.N. Trefethen, Computation of Pseudospectra, Acta Numerica, pages 247-295,(1999)
[10] L.N. Trefethen, L. Reichel, Eigenvalues and pseudoeigenvalues of Toeplitz matrices, Linear Algebra Appl., 162-164:143-185, 1992
[11] L.N. Trefethen, M. Embree, Spectra and Pseudospectra; The Behavior of Nonnormal Matrices and Operators, 2003 (a book in preparation).

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