Non-self-adjoint Eigenproblems

An -by- non-Hermitian matrix has eigenvalues, which can be anywhere in the complex plane. A non-self-adjoint regular pencil has from 0 to eigenvalues, which can be anywhere in the extended complex plane. Thus some of the choices in the self-adjoint case do not apply here. Instead, the following choices of spectral information are possible:

1. All the eigenvalues.
2. All the eigenvalues in some specified region of the complex plane, such as
   • The peripheral eigenvalues (when the spectrum is seen as a set in the complex plane), such as the 10 rightmost eigenvalues.
   • Eigenvalues in a given compact region, such as the unit disk.
   • Eigenvalues in a given unbounded region, such as the left half plane, or (a region around) the imaginary axis.

As before, the desired accuracy of the eigenvalues may be specified. For each of these choices, the user can also compute the corresponding (left or right) eigenvectors, or Schur vectors. Since the eigenvalues of a non-Hermitian matrix can be very ill-conditioned, it is sometimes hard to find all eigenvalues within a given region with certainty. For eigenvalues that are clustered together, the user may choose to estimate the mean of the cluster, or even the -pseudospectrum, the smallest region in the complex plane which contains all the eigenvalues of all matrices differing from the given matrix by at most : . The user may also choose to compute the associated invariant (or deflating or reducing) subspaces (left or right) instead of individual eigenvectors. However, due to the potential ill-conditioning of the eigenvalues, there is no guarantee that the invariant subspace will be well-conditioned.

A singular pencil has a more complicated eigenstructure, as defined by the Kronecker Canonical Form, a generalization of the Jordan Canonical Form [41][23]. Instead of invariant or deflating subspaces, we say a singular pencil has reducing subspaces.

Table 3 spells out the possibilities. In addition to the notation used in the last section, matrices denote generalized upper triangular (singular pencils only), matrices are generalized upper triangular in staircase form (singular pencils only), is in Jordan form, and is in Kronecker form. As before, these can be partial decompositions, when , , and are -by- instead of -by-.

  
Table 3: The Possible ``Eigendecompositions'' of Non-self-adjoint Eigenproblems

In addition to these decompositions, the user may request condition numbers for any of the computed quantities (eigenvalues, means of eigenvalue clusters, eigenvectors, invariant/deflating/reducing subspaces). Given computed values for eigenvalues, eigenvectors, and/or subspaces, the user may also request an a posteriori error bound based on a computed residual.