A Geometric Theory for the $QR$, $LU$ and Power Iterations

We are concerned with the task of computing the invariant subspaces of a given matrix. For this purpose the $LU$, $QR$, treppen and bi-iterations have been presented, used, and studied more or less independently of the old-fashioned power method. Each of these methods generates implicitly a sequence of subspaces which determines the convergence properties of the method. The iterations differ in the way in which a basis is constructed to represent each subspace. This aspect largely determines the usefulness of the method. We show that the first four iterations produce exactly the same sequence of subspaces as do direct and inverse iteration started from appropriate subspaces. Their convergence properties are therefore the same and we present a complete geometric convergence theory in terms of the power method. Most previous studies have been algebraic in character. We show that Hessenberg matrices are associated with ideal starting spaces. The theory rests naturally in the setting of an n-dimensional space $\mathcal{V}$ with no inner product structure. When $\mathcal{A}$ is a normal operator on an inner product space our results reduce to those of Buurema [3, Part I].