  This paper is to give a new understanding and applications of the subspace
projection method for selfadjoint eigenvalue problems. A new error estimate in
the energy norm, which is induced by the stiff matrix, of the subspace
projection method for eigenvalue problems is given. The relation between error
estimates in $L^2$-norm and energy norm is also deduced. Based on this
relation, a new type of inverse power method is designed for eigenvalue
problems and the corresponding convergence analysis is also provided. Then we
present the analysis of the geometric and algebraic multigrid methods for
eigenvalue problems based on the convergence result of the new inverse power
method.
