There are two different ways to get more than one linearly independent
eigenvector to a multiple eigenvalue, precisely as in the standard case.
The first is to restart and run
a projected operator, where all the converged
eigendirections have been projected
away; this corresponds to orthogonalizing the vector in step
8 of Algorithms 5.4 or 5.5
to the matrix
multiplied by all
converged eigenvectors. This procedure is repeated as long as new
vectors converge; see, e.g., [318].
We may also run a generalized variant of
block or band Lanczos, starting with several, say starting
directions, forming a block
, letting
operate on all of
in step
, to compute a new
-orthogonal block
. The matrix
will be a block tridiagonal, or more properly band matrix.
See a detailed description in [206].