Van der Waerden notation

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:

${\mathrm{\Lambda }}_{ϵ}(A)=\{\lambda \in \mathbb{ℂ}\mid \mathrm{\exists }x\in {\mathbb{ℂ}}^n\setminus \{0\},\mathrm{\exists }E\in {\mathbb{ℂ}}^{n\times n}:(A+E)x=\lambda x,\Vert E\Vert \le ϵ\}.$

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 [1]