Pseudospectrum

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 pseudospectrum of a matrix "A" for a given ε consists of all eigenvalues of matrices which are ε-close to "A"::Lambda_epsilon(A) = {lambda in mathbb{C} mid exists x in mathbb{C}^n, exists E in mathbb{C}^{n imes n} colon (A+E)x = lambda x, |E| leq epsilon }.

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".


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