$M= Q(\tan\Theta) U^{\mathsf{T}}$ is a condensed <span style="font-variant:small-caps;">svd</span> of the matrix $$M \coloneqq (I - AA^\mathsf{T})B(A^\mathsf{T} B)^{-1} \in \mathbb{R}^{n \times k}$$ and where $U \in \operatorname{O}(k)$ and $\Theta=\operatorname{diag}(\theta_1,\dots,\theta_k)\in \mathbb{R}^{k\times k}$ are as in and . Note that if $\cos\Theta=\Sigma$, then $\tan\Theta=(\Sigma^{-2}-I)^{1/2}$. Also, $\gamma(0) =\mathbf{A}$ and $\gamma(1) = \mathbf{B}$.

Aside from the Grassmann distance, there are many well-known distances between subspaces [@BN; @DD; @DHST; @EAS; @HamLee2008]. We present some of these in Table \[tab:distances\].

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                 *Principal angles*                                                                                                                *Orthonormal bases*
  Asimov         $d^{\alpha}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) =  \theta_k$                                                         $\cos^{-1} \lVert A^\mathsf{T} B \rVert_2$
  Binet–Cauchy   $d^{\beta}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \left(1 - \prod\nolimits_{i=1}^k\cos^2\theta_i\right)^{1/2}$        $(1 - (\det A^\mathsf{T} B)^2)^{1/2}$
  Chordal        $d^{\kappa}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \left(\sum\nolimits_{i=1}^k\sin^2\theta_i\right)^{1/2}$            $\frac{1}{\sqrt{2}} \lVert AA^\mathsf{T} - BB^\mathsf{T} \rVert_F$
  Fubini–Study   $d^{\phi}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \cos^{-1} \left(\prod\nolimits_{i=1}^k\cos \theta_i\right)$          $\cos^{-1} \lvert \det A^\mathsf{T} B \rvert$
  Martin         $d^{\mu}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \left( \log \prod\nolimits_{i=1}^k 1/\cos^2 \theta_i \right)^{1/2}$   $(-2 \log \det A^\mathsf{T} B)^{1/2}$
  Procrustes     $d^{\rho}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = 2\left(\sum\nolimits_{i=1}^k\sin^2(\theta_i/2)\right)^{1/2}$         $\lVert AU - BV \rVert_F$
  Projection     $d^{\pi}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = \sin \theta_k$                                                        $\lVert AA^\mathsf{T} - BB^\mathsf{T} \rVert_2$
  Spectral       $d^{\sigma}_{\operatorname{Gr}(k,n)}(\mathbf{A}, \mathbf{B}) = 2\sin (\theta_k/2)$                                                $\lVert AU - BV \rVert_2$
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The value $\sin \theta_1$ is sometimes called the *max correlation distance* [@HamLee2008] or spectral distance [@DHST] but it is not a distance in the 