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 &epsilon;-pseudospectrum of a matrix A consists of all eigenvalues of matrices which are &epsilon;-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.

More generally, for Banach spaces $$X,Y $$ and operators $$ A: X \to Y $$, one can define the $$ \epsilon$$-pseudospectrum of $$ A $$ (typically denoted by $$ \text{sp}_{\epsilon}(A) $$) in the following way
 * $$\text{sp}_{\epsilon}(A) = \{\lambda \in \mathbb{C} \mid \|(A-\lambda I)^{-1}\| \geq 1/\epsilon \}.$$

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