If the matrix to diagonalize is large and sparse, direct methods simply become impractical, also because many of the direct methods tend to destroy sparsity. As a result large dense matrices may arise during the diagonalization procedure. The idea behind iterative methods is to project the $n-$dimensional problem in smaller spaces, so-called Krylov subspaces. Given a matrix $\mathbf{A}$ and a vector $\mathbf{v}$, the associated Krylov sequences of vectors (and thereby subspaces) $\mathbf{v}$, $\mathbf{A}\mathbf{v}$, $\mathbf{A}^2\mathbf{v}$, $\mathbf{A}^3\mathbf{v},\dots$, represent successively larger Krylov subspaces.