Pseudospectra

Pseudospectra are similar (but not equal) to the eigenvalues of a matrix. Suppose A is a square matrix and λ is a scalar. We of course say that x is an eigenvector of A if Ax=λx where λ is the corresponding eigenvalue. The set of all eigenvalues of A is called the spectrum of A. We also say that the spectrum is the set of all λ such that det(A-λI)x=0. Eigenvalue analysis is used in a variety of applications throughout the mathematical sciences. Traditional methods rely on the fact that most matrices are normal. A normal matrix is a matrix with a complete set of orthogonal eigenvectors (e.g., symmetric and Hermitian matrices are normal).

When a matrix is nonnormal, eigenvalue analysis is not as straight-forward. This is where pseudospectra are useful. The pseudospectra (for a fixed ε) of a matrix A is the set of λ such that

For a normal matrix, the pseudospectra consist of closed ε-balls around each eigenvalue. For nonnormal matrices, the pseudospectra are larger. In applied mathematics, pseudospectra are useful because they often indicate much different behavior of the system than that indicated by the eigenvalues.

The picture above was generated with Matlab and corresponding EigTool package. The matrix used was a 10x10 companion matrix for the truncated power series of e^z. The black diamonds in the figure are the eigenvalues and the other curves define the pseudospectra.

More information on pseudospectra can be found at the Pseudospectra Gateway developed by Nick Trefethen and Mark Embree.