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:[1]
Λ\epsilon(A)=\{λ\inC\mid\existsx\inCn\setminus\{0\},\existsE\inCn\colon(A+E)x=λx,\|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.
More generally, for Banach spaces
X,Y
A:X\toY
\epsilon
A
sp\epsilon(A)
sp\epsilon(A)=\{λ\inC\mid\|(A-λI)-1\|\geq1/\epsilon\}.
\|(A-λI)-1\|=infty
A-λI
. Hogben. Leslie. Leslie Hogben . Handbook of Linear Algebra, Second Edition. 2013. CRC Press. 9781466507296. 23-1. 8 September 2017. en.