The following eigenproblems are typical, because they arise naturally in applications and because we have algorithms for them:
For each of these possibilities, the user can also compute a projection
of the matrix on the specified invariant subspace; if the subspace is
-dimensional, then the projection is a
by
matrix
whose eigendecomposition can be computed. The user can also compute the
right (and perhaps left) eigenvectors
in the computed invariant subspace.
For the eigenvalues that are clustered together,
the user may choose to compute the associated invariant subspace,
since in this case the individual eigenvectors can be very ill-conditioned,
while the invariant subspace may be less so.
Finally, for any of these quantities, the user might also want to
compute its condition number.
Even though we have effective algorithms for these problems,
we cannot necessarily solve all large scale problems
in an amount of time and space acceptable to all users.