Unitarily Invariant Metrics on the Grassmann Space

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.