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:

\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

Pseudospectra Gateway / Embree and Trefethen

Numerical linear algebra

Key concepts	Floating point Numerical stability

Problems	Matrix multiplication (algorithms) Matrix decompositions Linear equations Sparse problems

Hardware	CPU cache TLB Cache-oblivious algorithm SIMD Multiprocessing

Software	BLAS Specialized libraries General purpose software

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