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:
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 and operators , one can define the -pseudospectrum of (typically denoted by ) in the following way
where we use the convention that if is not invertible.
References
Bibliography Lloyd N. Trefethen and Mark Embree: "Spectra And Pseudospectra: The Behavior of Nonnormal Matrices And Operators", Princeton Univ. Press, (2005).
External links Pseudospectra Gateway by Embree and Trefethen