Let $L\_1$, $L\_2$ be two subspaces in $\mathbb{R}^n$ and $\dim(L\_1) = \dim(L\_2) = s<n$. 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.



Principal angles between subspaces and angle between unit vectors
$\newcommand\si\sigma\newcommand\al\alpha\newcommand\be\beta$Let $U:=L\_1$ and $V:=L\_2$. Without loss of generality, $s=\dim U=\dim V\ge1$. By the section [Angles between subspaces](https://en.wikipedia.org/wiki/Angles_between_flats#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 $\si+\al<s$ 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 $\angle(u,v)=\pi/2$. $\quad\Box$


