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:
Λ ϵ ( A ) = { λ ∈ C ∣ ∃ x ∈ C n ∖ { 0 } , ∃ E ∈ C n × n : ( A + E ) x = λ x , ‖ E ‖ ≤ ϵ } . {\displaystyle \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. More generally, for Banach spaces X , Y {\displaystyle X,Y} and operators A : X → Y {\displaystyle A:X\to Y} , one can define the ϵ {\displaystyle \epsilon } -pseudospectrum of A {\displaystyle A} (typically denoted by sp ϵ ( A ) {\displaystyle {\text{sp}}_{\epsilon }(A)} ) in the following way
sp ϵ ( A ) = { λ ∈ C ∣ ‖ ( A − λ I ) − 1 ‖ ≥ 1 / ϵ } . {\displaystyle {\text{sp}}_{\epsilon }(A)=\{\lambda \in \mathbb {C} \mid \|(A-\lambda I)^{-1}\|\geq 1/\epsilon \}.}
where we use the convention that ‖ ( A − λ I ) − 1 ‖ = ∞ {\displaystyle \|(A-\lambda I)^{-1}\|=\infty } if A − λ I {\displaystyle A-\lambda I} is not invertible.
References
Bibliography Lloyd N. Trefethen and Mark Embree: "Spectra And Pseudospectra: The Behavior of Nonnormal Matrices And Operators", Princeton Univ. Press, ISBN 978-0691119465 (2005).
External links Pseudospectra Gateway by Embree and Trefethen
