I am trying to understand Wedin Theorem on the perturbations of the Singular Vectors of a matrix, and a key element for this theorem is the matrix of the canonical angles between two subspaces; I am missing something about these matrices.
I have two bases: $$ \mathscr{V} = \{v_1,..,v_k\},\,\, and\,\,\mathscr{U} = \{u_1,..,u_k\}, $$ and the matrices whose columns are the vectors of those bases: $$ {V} = [v_1,..,v_k],\,\, and\,\,{U} = [u_1,..,u_k], $$
for two $k$ dimensional subspace of $\mathbb{R}^n$, $n>k$. I have two questions:
- Let $\Phi$ be the matrix of the canonical angles between the range of $\mathscr{V}$ and that of $\mathscr{U}$. What is the meaning of this matrix? Concretely, how can I calculate it?
- Assume that the Frobenius norm of $\sin \Phi$ is bounded, $$ ||\sin \Phi||_F < \epsilon $$ for some $\epsilon>0$. Can use this relation to create a bound on $||V^\top U||$? Or more, generally, how can I use this information to create a bound based on linear operations between matrices?
