Let $U, V \subset \mathbb{R}^n$ be two subspaces spanned by the columns of $A\in\mathbb{R}^{n\times m}$ and $B = A + E$, respectively. Moreover, let $n = 2m$, $A$ and $B$ have full rank, and $U \cap V = 0$ i.e. $U+V=\mathbb{R}^n$. Are there any lower bounds on the smallest principal angle (Miao et al., 1992) or, equivalently, upper bounds on its cosine, $$\cos(\theta_1)=\max_{\substack{u\in U, v\in V\\||u||=||v||=1}} u^\top v = ||P_U P_V||?$$ Here, $P_U, P_V$ denote the orthogonal projections onto $U$ and $V$ and $||\cdot||$ the spectral norm.
My attempt was to use that by Ipsen & Mayer (1995) the minimal angle $\theta_1$ between two complementary subspaces satisfies, $$\sin(\theta_1) = \dfrac{1}{||(P_U-P_V)^{-1}||}.$$ Hence, it would suffice to have a lower bound on the minimum singular value of $P_U-P_V$. However, I have been somewhat struggling to find results on this in the literature.
Thanks in advance for any help!
References
Miao, Jianming; Ben-Israel, Adi, On principal angles between subspaces in (\mathbb{R}^n), Linear Algebra Appl. 171, 81-98 (1992). ZBL0779.15003.
Sun, Jiguang, Perturbation of angles between linear subspaces, J. Comput. Math. 5, 58-61 (1987). ZBL0632.15009.
Ipsen, Ilse C. F.; Meyer, Carl D., The angle between complementary subspaces, Am. Math. Mon. 102, No. 10, 904-911 (1995). ZBL0842.15002.