In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, 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 operator

Normal operator

In mathematics, especially functional analysis, a normal operator on a complex Hilbert space H is a continuous linear operatorN:H\to Hthat commutes with its hermitian adjoint N*: N\,N^*=N^*N....

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:

Numerical algorithms which calculate the eigenvalues of a matrix

Eigenvalue algorithm

In linear algebra, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix. These eigenvalue algorithms may also find eigenvectors.-Characteristic polynomial:...

give only approximate results due to rounding and other errors. These errors can be described with the matrix E.