with a lower triangular matrix; see §10.3. The equivalent standard eigenvalue problem is

where and . As in Invert B, matrices like should never be evaluated explicitly, more in particular because they will in general not be sparse. For the application of an iterative method for the standard eigenvalue problem, we only need to provide the efficient evaluation of matrix-vector products, like

or

where and are given vectors. In practice, the vector can be formed in three steps:

The vector , when necessary, can be formed as follows:(a) solve for ,

(b) form ,

(c) solve for .

Since is a triangular matrix, the solutions of linear systems with or can be obtained by forward and backward substitutions. Sparsity can be exploited in a straightforward manner in all these steps.(a) solve for ,

(b) form ,

(c) solve for .