form of Krylov subspaces [@LS] and their variants [@C], as subspaces of structured matrices (e.g. Toeplitz, Hankel, banded), and in recent developments such as compressive sensing (e.g. Grassmannian dictionaries [@SH], online matrix completion [@BNR]).

One of the most basic problems with subspaces is to define a notion of separation between them. The solution is well-known for subspaces of the same dimension $k$ in $\mathbb{R}^n$. These are points on the Grassmannian $\operatorname{Gr}(k,n)$, a Riemannian manifold, and the geodesic distance between them gives us an intrinsic distance. The *Grassmann distance* is independent of the choice of coordinates and can be readily related to principal angles and thus computed via the singular value decomposition (<span style="font-variant:small-caps;">svd</span>): For subspaces $\mathbf{A}, \mathbf{B} \in \operatorname{Gr}(k,n)$, form matrices $A, B \in \mathbb{R}^{n \times k}$ whose columns are their respective orthonormal bases, then $$\label{eq:grassdist}
d(\mathbf{A},\mathbf{B}) = \Bigl(\sum\nolimits_{i=1}^k \theta_i^2\Bigr)^{1/2},$$ where $\theta_i = \cos^{-1} \bigl(\sigma_i(A^\mathsf{T} B)\bigr)$ is the $i$th principal angle between $\mathbf{A}$ and $\mathbf{B}$. This is the geodesic distance on the Grassmannian viewed as a Riemannian manifold. There are many other common distances defined on Grassmannians — Asimov, Binet–Cauchy, chordal, Fubini–Study, Martin, Procrustes, projection, spectral (see Table \[tab:distances\]).

What if the subspaces are of different dimensions? In fact, if 