- Source: 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:
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.
Notes
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