Consider the maximization problem for a $p \times p$ covariance matrix $S$: $$\max_{\Theta \succ 0} \log \det \Theta - tr(S\Theta)-\lambda \sum_{i\neq j} |\Theta_{ij}|$$ I need to show that for integers $0=p_0 < p_1 < ... < p_K = p$ and $C_j = \{i: p_{j-1}+1 \leq i \leq p_j \}$, the optimal solution $\Theta^*_{ij} = 0$ for any $(i,j) \notin \bigcup_{l=1}^K (C_l \times C_l)$ i.e. $\Theta^*$ is a block diagonal matrix of the form: $$\Theta^* = \begin{pmatrix} \Theta^*_1 & & \\ & \ddots & \\ & & \Theta^*_K \end{pmatrix}$$ with $\Theta^*_k$ being a $|C_k| \times |C_k|$ matrix for $k=1,..,K$ if and only if $|S_{ij}| \leq \lambda$ for any $(i,j) \notin \bigcup_{l=1}^K (C_l \times C_l) $
I must admit I am completely lost on this problem and don't even know where to begin. Any hints/suggestions/references would be much appreciated!