Pseudospectrum

In mathematics, the pseudospectrum of an operator is a set containing the spectrum of the operator and the numbers that are "almost" eigenvalues. Knowledge of the pseudospectrum can be particularly useful for understanding non-normal operators and their eigenfunctions. The ε-pseudospectrum of a matrix A consists of all eigenvalues of matrices which are ε-close to A:

${\mathrm{\Lambda }}_{ϵ}(A)=\{\lambda \in \mathbb{ℂ}\mid \mathrm{\exists }x\in {\mathbb{ℂ}}^n\setminus \{0\},\mathrm{\exists }E\in {\mathbb{ℂ}}^{n\times n}:(A+E)x=\lambda x,\Vert E\Vert \le ϵ\}.$

Numerical algorithms which calculate the eigenvalues of a matrix give only approximate results due to rounding and other errors. These errors can be described with the matrix E.

More generally, for Banach spaces $X,Y$ and operators $A:X\to Y$ , one can define the $ϵ$-pseudospectrum of $A$ (typically denoted by ${\text{sp}}_{ϵ}(A)$) in the following way

${\text{sp}}_{ϵ}(A)=\{\lambda \in \mathbb{ℂ}\mid \Vert (A-\lambda I{)}^{-1}\Vert \ge 1/ϵ\}.$

where we use the convention that $\Vert (A-\lambda I{)}^{-1}\Vert =\mathrm{\infty }$ if $A-\lambda I$ is not invertible.

Bibliography

• Lloyd N. Trefethen and Mark Embree: "Spectra And Pseudospectra: The Behavior of Nonnormal Matrices And Operators", Princeton Univ. Press, ISBN:978-0691119465 (2005).

External links

• Pseudospectra Gateway by Embree and Trefethen