Let a quantum system be described by a correlation matrix $$ C_{ij} = \langle c_i^\dagger c_j \rangle\ , $$ which we can split in components $A$ and $\bar{A}$. I have read that we can calculate the entanglement entropy between these components by restricting the correlation matrix to the rows and columns corresponding to $A$, calculating its eigenvalues $\{\nu_p\}$, and then $$ S = -\sum_p \left( \nu_p \log \nu_p + (1-\nu_p)\log(1-\nu_p) \right)\ . $$
I'm trying to find any source where this statement is proven. I would appreciate if anyone can share any source about this.