Abstract
Let ${\cal G}_{m,n}$ be the Grassmann space of m-dimensional subspaces of $\mathbb{F}^{n}$. Denote by $\theta_{1}({{\cal X}, {\cal Y}}), \ldots,\theta_{m}({{\cal X},{\cal Y}})$ the canonical angles between subspaces ${{\cal X}}, {\cal Y} \in \cG_{m,n}$. It is shown that $\Phi(\theta_{1}({{\cal X}, {\cal Y}}),\ldots,\theta_{m}({\cal X}, {\cal Y}))$ defines a unitarily invariant metric on ${\cal G}_{m,n}$ for every symmetric gauge function $\Phi$. This provides a wide class of new metrics on ${\cal G}_{m,n}$. Some related results on perturbation and approximation of subspaces in ${\cal G}_{m,n}$, as well as the canonical angles between them, are also discussed. Furthermore, the equality cases of the triangle inequalities for several unitarily invariant metrics are analyzed.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have