Pseudospectrum Explained

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

and operators

A:X\toY

, one can define the

\epsilon

-pseudospectrum of

A

(typically denoted by

sp\epsilon(A)

) in the following way

sp\epsilon(A)=\{λ\inC\mid\|(AI)-1\|\geq1/\epsilon\}.

where we use the convention that

\|(AI)-1\|=infty

if

A-λI

is not invertible.[2]

Bibliography

External links

Notes and References

  1. Book: Leslie Hogben

    . Hogben. Leslie. Leslie Hogben . Handbook of Linear Algebra, Second Edition. 2013. CRC Press. 9781466507296. 23-1. 8 September 2017. en.

  2. Book: Böttcher. Albrecht. Silbermann. Bernd. Albrecht Böttcher. Introduction to Large Truncated Toeplitz Matrices. 1999. Springer New York. 978-1-4612-1426-7. 70. 10.1007/978-1-4612-1426-7_3. en.