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.
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The pseudospectrum of a matrix ''A'' for a given &epsilon; consists of all eigenvalues of matrices which are &epsilon;-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}}

Latest revision as of 16:43, 14 December 2014

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