Pseudospectrum

Summary

The $\varepsilon$-pseudospectrum of a square matrix is a set of complex numbers. It consists of the numbers that become eigenvalues after some perturbation of the matrix of norm less than $\varepsilon$. Equivalently, it is the region where the norm of the resolvent $(zI - A)^{-1}$ exceeds $1/\varepsilon$. For non-normal matrices it can be much larger than the set of eigenvalues. It then explains transient growth, slow convergence of iterative methods and sensitivity of computed eigenvalues, which the eigenvalues alone do not show.