I'm currently wondering about a possible continuation of my previous question: Vector Analysis & Linear Algebra [all of these question are not part of any homework assignment. I'm just trying to understand an article, which uses the inequalities I'm asking about]
We know that there exist vectors $b_i, b_j$ such that $\angle (u,b_j) \leq \dfrac{\delta}{2} $ and $ |b_i | \leq \dfrac{\delta}{4} |b_j| $ .
We are now given two matrices $B_j, B_i \in O(n)$ such that $\max \left\{ \dfrac{|B_jB_i^{-1}x-x|}{|x|} \right\} \leq \epsilon $.
Why does this imply that the vector $b:= b_j - B_jB_i^{-1}b_i$ satisfies $\angle (u,b) \leq \delta$ ?
Thanks !
