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Here are several examples showing pseudospectra for particular classes of matrices. These examples only begin to illustrate the many interesting operators for which pseudospectra have been investigated. Other important classes include non-self-adjoint differential operators, applications in fluid mechanics, and various spectral discretizations. For a more extensive list, see the catalog of Applications, or try clicking on the thumbnail images at the top of the home page.

Toeplitz Matrices
and Operators

Toeplitz matrices provide one of the most compelling applications of pseudospectra. A Toeplitz matrix has constant entries on each diagonal, and the corresponding infinite-dimensional Toeplitz operator is a singly-infinite matrix. The constants on the diagonals are the Laurent coefficients of the symbol, a complex-valued function whose domain is the unit circle, T.

The spectrum of the Toeplitz operator is determined by the symbol, a. If a is continuous, then the spectrum is the a(T) together with all points this curve encloses with non-zero winding number. The eigenvalues of finite Toeplitz matrices are very different, typically falling on curves in the complex plane for arbitrarily large (but finite) dimensions. It is well known that these eigenvalues are typically difficult to compute accurately.

It turns out that while the eigenvalues of finite Toeplitz matrices do not generally converge to the spectrum of the corresponding infinite-dimensional operator, the pseudospectra of the finite matrices do converge to the operator pseudospectra for a broad class of symbols. In the pages linked to below, we illustrate different aspects of this convergence.

Random Matrices

Random matrices are of importance in a wide variety of applications, and in many instances these matrices are non-normal. When the degree of non-normality is large, pseudospectra can help explain interesting phenomena. The links below show a number of illustrations.

Many random matrices can be classified as "stochastic Toeplitz" matrices; that is, the entries on any one diagonal are all independent samples from the same probability distribution. (Some diagonals may be constant, corresponding to a distribution that is a delta function.)

In this section, we illustrate spectra and pseudospectra for several specific stochastic Toeplitz matrices.