Van der Waerden notation: Difference between revisions
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In [[mathematics]], the '''pseudospectrum''' of an [[Operator (mathematics)|operator]] is a [[Set (mathematics)|set]] containing the [[spectrum of an operator|spectrum]] of the operator and the numbers that are "almost" eigenvalues. Knowledge of the pseudospectrum can be particularly useful for understanding [[normal operator|non-normal operator]]s and their eigenfunctions. | |||
The pseudospectrum of a matrix ''A'' for a given ε consists of all eigenvalues of matrices which are ε-close to ''A'': | |||
:<math>\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> | |||
[[Eigenvalue algorithm|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''. | |||
==See also== | |||
[[Pseudo-spectral method]] | |||
== References == | |||
* Pseudospectra Gateway / Embree and Trefethen [http://web.comlab.ox.ac.uk/pseudospectra/] | |||
[[Category:Numerical linear algebra]] | |||
{{Numerical linear algebra}} | |||
Revision as of 14:27, 5 June 2013
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 pseudospectrum of a matrix A for a given ε consists of all eigenvalues of matrices which are ε-close to A:
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.
See also
References
- Pseudospectra Gateway / Embree and Trefethen [1]