\times l}$ and $\Theta_1 = \operatorname{diag}(\theta_1,\dots,\theta_{r}) \in \mathbb{R}^{r \times r}$. Consider the column vectors, $$AU = [p_1,\dots,p_k], \quad BV = [q_1,\dots,q_l].$$ The principal vectors are given by $(p_1,q_1),\dots,(p_r, q_r)$. Strictly speaking, principal vectors come in pairs but we will also call the vectors $p_{r+1},\dots,p_k$ (if $r = l < k$) or $q_{r+1},\dots,q_l$ (if $r = k < l$) principal vectors for lack of a better term.

We will use the following fact from [@GVL Theorem 6.4.2].

\[prop:angles\] Let $r = \min (k,l)$ and $\theta_1,\dots,\theta_{r}$ and $(p_1,q_1),\dots,(p_{r}, q_{r})$ be the principal angles and principal vectors between $\mathbf{A} \in \operatorname{Gr}(k,n)$ and $\mathbf{B} \in \operatorname{Gr}(l,n)$ respectively. If $m < r$ is such that $1 = \cos \theta_1 = \dots = \cos \theta_m > \cos \theta_{m+1}$, then $$\mathbf{A} \cap \mathbf{B} = \operatorname{span} \{p_1,\dots, p_m \} =  \operatorname{span} \{q_1,\dots, q_m \}.$$

If $k = l$, the geodesic distance between $\mathbf{A}$ and $\mathbf{B}$ in $\operatorname{Gr}(k,n)$ is called the *Grassmann distance* and is given by $$\label{eq:grassdist1}
d_{\operatorname{Gr}(k,n)}(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^k \theta_i^2\Bigr)^{1/2} = \lVert\cos^{-1}\Sigma\rVert_{F}.$$ An explicit expression for the geodesic [@AMS] connecting $\mathbf{A}$ to $\mathbf{B}$ on $\operatorname{Gr}(k,n)$ that minimizes the Grassmann distance is given by $\gamma : [0,1] \to \operatorname{Gr}(k,n)$, $$\label{eq:geodesic}
\gamma(t) = \operatorname{span}( AU\cos t\Theta+ Q\sin t \Theta )$$ where 