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:

 1. 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?
 2. 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?