# Principal angles between subspaces and angle between unit vectors

Let $$L_1$$, $$L_2$$ be two subspaces in $$\mathbb{R}^n$$ and $$\dim(L_1) = \dim(L_2) = s. Let $$0 \leq \theta_1 \leq \dotsb\leq \theta_{s} \leq \pi/2$$ be the principal angles between $$L_1, L_2$$. Let there be two unit vectors $$u_1\in L_1$$ and $$u_2 \in L_2$$ and the principal angle between $$\operatorname{span}\{u_1\}$$ and $$\operatorname{span}\{u_2\}$$ is $$\alpha \in [0,\pi/2]$$. Is there any known upper bound on $$\alpha$$ in terms of $$\theta_{s}$$?

Here is my try: Let $$U_1, U_2 \in \mathbb{R}^{n \times s}$$ be two orthogonal matrices such that their columns are orthonormal bases of $$L_1$$ and $$L_2$$ respectively. Now any unit vector in $$L_j$$ is essentially of the form $$U_j \alpha_j$$ where $$\lVert \alpha_j\rVert_2 =1$$ for $$j \in \{1,2\}$$. Also, the singular values of $$U_1^\top U_2$$ are $$\{\cos \theta_1, \dotsc, \cos \theta_s\}$$. Now, I don't have any idea how to proceed any further. Any help will be appreciated.

$$\newcommand\si\sigma\newcommand\al\alpha\newcommand\be\beta$$Let $$U:=L_1$$ and $$V:=L_2$$. Without loss of generality, $$s=\dim U=\dim V\ge1$$. By the section Angles between subspaces of the Wikipedia article, there are orthonormal bases $$(a_1,\dots,a_s)$$ and $$(b_1,\dots,b_s)$$ of $$U$$ and $$V$$, respectively; nonnegative integers $$\si$$ and $$\al$$; and $$\be_1,\dots,\be_\al$$ in the interval $$(0,\pi/2)$$ such that for any $$u\in U$$ and $$v\in V$$ we have $$u\cdot v=\sum_{i=1}^\si u^i v^i+\sum_{i=\si+1}^{\si+\al} u^i v^i\cos\be_i,$$ where $$u\cdot v$$ is the dot product of $$u$$ and $$v$$; the $$u^i$$'s are the coordinates of $$u$$ in the basis $$(a_1,\dots,a_s)$$; and the $$v^i$$'s are the coordinates of $$v$$ in the basis $$(b_1,\dots,b_s)$$.
So, if $$\si+\al or $$\si+\al>1$$, then $$u\cdot v=0$$ and hence $$\angle(u,v)=\pi/2$$ for some nonzero $$u\in U$$ and $$v\in V$$.
Otherwise, if $$\si+\al\ge s$$ and $$\si+\al\le1$$, then $$s=1$$.
Thus, excluding the trivial case $$s=1$$, the maximal angle between $$\text{span}\{u\}$$ and $$\text{span}\{v\}$$ for nonzero vectors $$u\in U$$ and $$v\in V$$ is always $$\pi/2$$.
This conclusion can also be obtained more elementarily, as follows. Suppose that $$s\ge2$$ and let $$(a_1,\dots,a_s)$$ be any basis of $$U$$. Take any nonzero $$v\in V$$. Then for for some nonzero $$(u^1,\dots,u^s)\in\mathbb R^s$$ and $$u:=\sum_{i=1}^s u^i a_i$$ we have $$u\ne0$$ and $$u\cdot v=\sum_{i=1}^s u^i(a_i\cdot v)=0$$ and hence $$\angle(u,v)=\pi/2$$. $$\quad\Box$$