Approximately invariant subspaces

Invariant subspaces are well documented in the literature and approximations for them exist. Approximately invariant subspaces have properties that are highly desirable for iterative solution strategies of large sparse matrix systems and for approximating Ritz values and Ritz vectors of such matrices. It is often a difficult task to identify an approximately invariant subspace numerically. In this work a new definition is proposed that assists with the task of identifying when a subspace is approximately invariant by measuring the sine of the angle between the image of any vector in the subspace and its orthogonal projection onto the subspace. In particular the effect that different bases have on this measure is analysed. Finally, the definition is used to provide theoretical error estimates when solving either systems of equations or the eigenvalue problem.