Multiple Eigenvalues.

As stated in the previous paragraph, we would get convergence only to eigenvectors that were represented in the starting vector. This is important when the matrix pencil (5.1) has multiple eigenvalues. In that case, we will get only one vector from the corresponding multidimensional invariant subspace.

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 $r$ in step 8 of Algorithms 5.4 or 5.5 to the matrix $B$ 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 $p$ starting directions, forming a block $V_1$, letting $A$ operate on all of $V_j$ in step $j$, to compute a new $B$-orthogonal block $V_{j+1}$. The matrix $T$ will be a block tridiagonal, or more properly band matrix. See a detailed description in [206].