function defined by $$\delta(\mathbf{A},\mathbf{B})=\Bigl(\sum\nolimits_{i=1}^k\theta_i^2\Bigr)^{1/2}$$ where $\theta_i \coloneqq \theta_i(\mathbf{A},\mathbf{B})$, $i=1,\dots,k$. Then $$\delta(\mathbf{A},\mathbf{B})\ge d_{\operatorname{Gr}(l,n)}\bigl(\mathbf{B},\Omega_{+}(\mathbf{A})\bigr).$$

It suffices to find an $\mathbf{X}\in \Omega_{+}(\mathbf{A})$ such that $\delta(\mathbf{A},\mathbf{B})=d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})$. Let $(p_1,q_1),\dots, (p_k,q_k)$ be the principal vectors between $\mathbf{A}$ and $\mathbf{B}$. We will extend $q_1,\dots,q_k$ into an orthonormal basis of $\mathbf{B}$ by appending appropriate orthonormal vectors $q_{k+1},\dots,q_l$. The principal angles are given by $\theta_i=\cos^{-1} p_i^\mathsf{T} q_i$, $\lVert p_i \rVert =\lVert q_i \rVert=1$. If we take $\mathbf{X} \in \operatorname{Gr}(l,n)$ to be the subspace spanned by $p_1,\dots,p_k,q_{k+1},\dots, q_l$, then $$\begin{aligned}
\label{eq:ineq1}
d_{\operatorname{Gr}(l,n)}(\mathbf{X},\mathbf{B})
&= [(\cos^{-1} p_1^\mathsf{T} q_1)^2 + \dots +  (\cos^{-1} p_k^\mathsf{T} q_k)^2 \nonumber \\
&\qquad  +  (\cos^{-1}  q_{k+1}^\mathsf{T} q_{k+1})^2 + \dots + (\cos^{-1}  q_l^\mathsf{T} q_l)^2 ]^{1/2}\\
&= [ \theta_1^2 + \dots +  \theta_k^2 + 0^2 + \dots + 0^2]^{1/2} = \delta(\mathbf{A},\mathbf{B}). \nonumber \end{aligned}$$

We state the following well-known fact [@Horn Corollary 3.1.3] for easy reference and deduce a corollary that will be useful for Lemma \[lem:ineq2\].

\[prop:compare\] Let $k\le l\le n$ be positive integers. Suppose $B \in \mathbb{R}^{n\times l}$ and $B_k \in \mathbb{R}^{n\times k}$ is a submatrix obtained by removing any $l-k$ columns from $B$. Then the $i$th singular values satisfy $\sigma_i(B_k)\le \sigma_i(B)$ for $i=1,\dots, k$.

\[cor:compare\] Let $B$ and $B_k$ be as in Proposition \[prop:compare\] and $\mathbf{B}$ and $\mathbf{B}_k$ 