# 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]

## Notes

1. Hogben, Leslie (2013) (in en). Handbook of Linear Algebra, Second Edition. CRC Press. p. 23-1. ISBN 9781466507296. Retrieved 8 September 2017.
2. Böttcher, Albrecht; Silbermann, Bernd (1999) (in en). Introduction to Large Truncated Toeplitz Matrices. Springer New York. p. 70. ISBN 978-1-4612-1426-7. Retrieved 22 March 2022.

## Bibliography

• Lloyd N. Trefethen and Mark Embree: "Spectra And Pseudospectra: The Behavior of Nonnormal Matrices And Operators", Princeton Univ. Press, ISBN:978-0691119465 (2005).