# Pseudospectra.jl

Pseudospectra is a Julia package for computing pseudospectra of non-symmetric matrices, and plotting them along with eigenvalues ("spectral portraits"). Some related computations and plots are also provided.

## Introduction

Whereas the spectrum of a matrix is the set of its eigenvalues, a pseudospectrum is the set of complex numbers "close" to the spectrum in some practical sense.

More precisely, the ϵ-pseudospectrum of a matrix A, $\sigma_{\epsilon}(A)$, is the set of complex numbers $\lambda$ such that

• $\lambda$ is an eigenvalue of some matrix $A+E$, where the perturbation $E$ is small: $\|E\| < \epsilon$
• the resolvent at $\lambda$ has a large norm: $\|(A-λI)^{-1}\| > 1/\epsilon$,

(the definitions are equivalent). Specifically, this package is currently limited to the unweighted 2-norm.

Among other things, pseudospectra:

• elucidate transient behavior hidden to eigen-analysis, and
• indicate the utility of eigenvalues extracted via iterative methods like eigs.

This package facilitates computation, display, and investigation of the pseudospectra of matrices and some other representations of linear operators.

## Spectral portraits

It is customary to display pseudospectra as contour plots of the logarithm of the inverse of the resolvent norm $\epsilon = 1/\|(A-zI)^{-1}\|$ for $z$ in a subset of the complex plane. Thus $\sigma_{\epsilon}(A)$ is the union of the interiors of such contours. Such plots, sometimes called spectral portraits, are the most prominent product of this package.

The figure shows a section of the complex plane with eigenvalues and contours of log10(ϵ). It was generated by the following code:

using Plots, Pseudospectra, LinearAlgebra
n=150
B=diagm(1 => fill(2im,n-1), 2 => fill(-1,n-2), 3 => fill(2,n-3), -2 => fill(-4,n-2), -3 => fill(-2im, n-3))
spectralportrait(B)

## Credit

Pseudospectra.jl is largely a translation of the acclaimed MATLAB-based EigTool (homepage here)

## References

• The Pseudospectra gateway.
• L.N. Trefethen and M.Embree, Spectra and Pseudospectra; The Behavior of Nonnormal Matrices and Operators, Princeton 2005,