where . Alternatively, the problem (8.1) is also equivalent to

where . If the right eigenvectors are of primary interest, then (8.3) is preferable, since it avoids an additional back transformation.

For an iterative method, as described in Chapter 7,
for the reduced standard eigenvalue problem (8.3)
or (8.4), it is not necessary to evaluate
the product or . This is a key observation
in the treatment of large sparse and .
One only needs to evaluate matrix-vector products, like

for a given vector . For a two-sided Lanczos-type method, the vector

is also necessary for given vector . Note that the vector can be computed in two steps:

Similarly, the vector , if necessary, can be evaluated as(a) form ,

(b) solve for .

One can exploit sparsity in each of these steps. In general, the iterative methods, with the exception of the Jacobi-Davidson method, require accurate solution of the linear systems at step (b) (and (b)). If possible, a direct linear system solver, say, using LU factorization of , is preferable. See §10.3 for the discussions on dense or sparse LU factorization.(a) solve for ,

(b) form .

The error introduced by this transformation to standard form can be proportional to . If is ill-conditioned, then the approach is potentially suspect. In that situation, one may consider the SI for the transformation or the usage of the Jacobi-Davidson method discussed below.