Let $\bf A$ be a symmetric matrix and let $\Omega_k$ denote the set of symmetric positive semidefinite matrices (SPSD) with rank at most $k$. Consider the following optimization problem
$$ \arg \min_{{\bf X} \in \Omega_k} \frac{1}{2} \left\| {\bf X} − {\bf A} \right\|_{\text{F}}^{2} $$
where $ {\left\| \cdot \right\|}_{\text{F}} $ is the Frobenius norm.
I know that the minimizer can be computed using the spectral decomposition of $\bf A$. What I am not sure about is how to go about the proof. It would be great if someone could give a sketch of the proof (or suggest a reference).