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:

$\Lambda_\epsilon(A) = \{\lambda \in \mathbb{C} \mid \exists x \in \mathbb{C}^n \setminus \{0\}, \exists E \in \mathbb{C}^{n \times n} \colon (A+E)x = \lambda x, \|E\| \leq \epsilon \}.$

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.

References

• Pseudospectra Gateway / Embree and Trefethen [1]