# The principal angles and the gap

Taslaman, Leo (2014) The principal angles and the gap. [MIMS Preprint]

In this note we provide proofs for some known results on the principal angles and the gap between two subspaces of $C^n$. Both the principal angles and the gap are introduced with respect to an arbitrary positive definite inner product. We show that the principal angles between two subspaces $U$ and $V$ are unique and prove that the largest one, $\theta_{\max}$, satisfies $\theta_{\max} = \max_{u\in U, \|u\|=1} \min_{v\in V, \|v\|=1} \angle(u,v)$ and $\sin\theta_{\max} =gap(U,V)$ when $\dim U=\dim V$.