If a matrix $\bf X$ is factored into a low dimensional matrix such that $\bf X = UV'$, it says
$$ \min_{~~~\bf U,V \\X=UV'} \bf \|U\| \|V\| = \min_{~~~\bf U,V \\X=UV'} \frac{ \|U\|^2 + \|V\|^2}{2} = \sum_i \sigma_i $$
where $\sigma_i$ are the singular values of the matrix $\bf X$ and $\|\circ\|$ is the Frobenius norm. This looks like a very standard result, where can I see the proof for this ?