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). Handbook of Linear Algebra, Second Edition. CRC Press. p. 23-1. ISBN 9781466507296. Retrieved 8 September 2017.
2. Böttcher, Albrecht; Silbermann, Bernd (1999). Introduction to Large Truncated Toeplitz Matrices. Springer New York. p. 70. doi:10.1007/978-1-4612-1426-7_3. ISBN 978-1-4612-1426-7.

Bibliography

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

External links

• Pseudospectra Gateway by Embree and Trefethen