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]

\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

^ Hogben, Leslie (2013). Handbook of Linear Algebra, Second Edition. CRC Press. p. 23-1. ISBN 9781466507296. Retrieved 8 September 2017.

Lloyd N. Trefethen and Mark Embree: "Spectra And Pseudospectra: The Behavior of Nonnormal Matrices And Operators", Princeton Univ. Press, ISBN 978-0691119465 (2005).
Pseudospectra Gateway / Embree and Trefethen [1]

[Hogben-1] Hogben, Leslie (2013). Handbook of Linear Algebra, Second Edition. CRC Press. p. 23-1. ISBN 9781466507296. Retrieved 8 September 2017.

[1]