The difference is that we do not need to

diagonalize [S.

For any system except for the very smallest (less than about 1000 atoms), iterative techniques, such as the Arnoldi method (10), must be used to

diagonalize the system Hamiltonian and only those eigenstates near the fundamental gap are found.

It has the advantage of not having to

diagonalize or block

diagonalize A at the disadvantage of a computational complexity beyond O([n.

For the evolution of the energy level, it is a necessary to

diagonalize the Hamiltonian [[?

Conversely, we can

diagonalize the curved metric at any point, such as to reduce it to a locally flat metric (tangent space), but the flat coordinates (axes) will not, in general, be the same for all the points; they depend, in general, on the point.

Although one can use a mass lumping procedure to

diagonalize the computed mass matrix, the projection property is not valid and the super-convergence property is, in general, lost after doing the mass lumping procedure.

On the other hand, the zero Dirichlet BCs case is a bit more involved since we cannot

diagonalize the related matrix in general: moreover, under the assumption of Dirichlet BCs, a pure Toeplitz structure arises and indeed a similar analysis can be carried out by using the symbol (see [ 16]).

The new idea is simple: we use approximate right and left eigenvectors obtained from the ULA to build a projected matrix which we

diagonalize using a standard method.

Therefore to complete our task we need to

diagonalize the operator H, i.