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., .
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 .