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:^[1]

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

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 ${\displaystyle X,Y}$ and operators ${\displaystyle A:X\to Y}$ , one can define the ${\displaystyle \epsilon }$-pseudospectrum of ${\displaystyle A}$ (typically denoted by ${\displaystyle {\text{sp}}_{\epsilon }(A)}$) in the following way

${\displaystyle {\text{sp}}_{\epsilon }(A)=\{\lambda \in \mathbb {C} \mid \|(A-\lambda I)^{-1}\|\geq 1/\epsilon \}.}$

where we use the convention that ${\displaystyle \|(A-\lambda I)^{-1}\|=\infty }$ if ${\displaystyle A-\lambda I}$ is not invertible.^[2]