If I have the two unitary matrices from the SVD of an m x n matrix (U, V*) and I form a set of new matrices by doing $u_iv_i^H$ (forms an m x n matrix). Assuming $r = min(m, n)$ and my set is $X_1, X_2, ..., X_r$, how can I show that this is an orthonormal set? Is there some property of unitary matrices that I've forgotten? Their rows and columns form an orthonormal basis in $C^n$ but what of the product of two orthonormal vectors?
Furthermore, how do I generalize the coordinates of the original matrix using this orthonormal basis of the vector space?