# The principal angles and the gap

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

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## Abstract

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 $\dim U=\dim V$.

Item Type: MIMS Preprint principal angles, canonical angles, gap, canonical correlations MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory Leo Taslaman 03 Mar 2014 20 Oct 2017 14:13 http://eprints.maths.manchester.ac.uk/id/eprint/2105

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