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]
- [math]\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 \}. }[/math]
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 [math]\displaystyle{ X,Y }[/math] and operators [math]\displaystyle{ A: X \to Y }[/math] , one can define the [math]\displaystyle{ \epsilon }[/math]-pseudospectrum of [math]\displaystyle{ A }[/math] (typically denoted by [math]\displaystyle{ \text{sp}_{\epsilon}(A) }[/math]) in the following way
- [math]\displaystyle{ \text{sp}_{\epsilon}(A) = \{\lambda \in \mathbb{C} \mid \|(A-\lambda I)^{-1}\| \geq 1/\epsilon \}. }[/math]
where we use the convention that [math]\displaystyle{ \|(A-\lambda I)^{-1}\| = \infty }[/math] if [math]\displaystyle{ A - \lambda I }[/math] is not invertible.[2]
Notes
- ↑ Hogben, Leslie (2013) (in en). Handbook of Linear Algebra, Second Edition. CRC Press. p. 23-1. ISBN 9781466507296. https://books.google.com/books?id=Er7MBQAAQBAJ&dq=pseudospectrum&pg=SA23-PA18. Retrieved 8 September 2017.
- ↑ Böttcher, Albrecht; Silbermann, Bernd (1999) (in en). Introduction to Large Truncated Toeplitz Matrices. Springer New York. p. 70. ISBN 978-1-4612-1426-7. https://doi.org/10.1007/978-1-4612-1426-7_3. 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).
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