Unitarily invariant metrics on the grassmann space

Let G _m,n be the Grassmann space of m-dimensional subspaces of double-struck F sign ⁿ. Denote by θ ₁(X,y), . . . ,θ _m(X,y) the canonical angles between subspaces X,y ∈ G _m,n. It is shown that ψ(θ ₁(X,y), . . . ,θ _m(X,y)) defines a unitarily invariant metric on G _m,n for every symmetric gauge function ψ. This provides a wide class of new metrics on G _m,n. Some related results on perturbation and approximation of subspaces in 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.