${\bf M}$ is NOT Hermitian Positive Semi-Definite

 If neither ${\bf A}$ nor ${\bf M}$ is Hermitian positive semi-definite, then a direct transformation to standard form is required. One simple way to obtain a direct transformation of equation (3.2.1) to a standard eigenvalue problem is to multiply on the left by which results in Of course, one should not perform this transformation explicitly since it will most likely convert a sparse problem into a dense one. If possible, one should obtain a direct factorization of ${\bf M}$ and when a matrix-vector product involving is called for, it may be accomplished with the following two steps:

1.

Matrix-vector multiply

2.

Solve:

Several problem dependent issues may modify this strategy. If ${\bf M}$is singular or if one is interested in eigenvalues near a point then a user may choose to work with but without using the ${\bf M}$-inner products discussed previously. In this case the user will have to transform the converged eigenvalues of to eigenvalues of the original problem.