${\bf M}$ is Hermitian Positive Definite

 If ${\bf M}$ is Hermitian positive definite and well conditioned ( is of modest size), then computing the Cholesky factorization   and converting equation (3.2.1) to

provides a transformation to a standard eigenvalue problem.   In this case, a request for a matrix vector product would be satisfied with the following three steps:

1.

Solve for

2.

Matrix-vector multiply

3.

Solve for

Upon convergence, a computed eigenvector for is converted to an eigenvector of the original problem by solving the the triangular system This transformation is most appropriate when ${\bf A}$ is Hermitian, ${\bf M}$ is Hermitian positive definite and extremal eigenvalues are sought. This is because will be Hermitian when ${\bf A}$ is.

If ${\bf A}$ is Hermitian positive definite and the smallest eigenvalues are sought, then it would be best to reverse the roles of ${\bf A}$ and ${\bf M}$ in the above description and ask for the largest algebraic eigenvalues or those of largest magnitude. Upon convergence, a computed eigenvalue would then be converted to an eigenvalue of the original problem by the relation