I have a class of symmetric block matrices of the form
$$\begin{pmatrix}D_1 \ B\\B^T \ D_2\end{pmatrix}$$
where $D_1,D_2$ are diagonal matrices with positive diagonal entries and $B$ is a (non-square) matrix with nonpositive entries. $B$ is pretty sparse with exactly three negative entries in each row, the rest being zero. The size and concrete makeup of the matrix is variable, although there are some known relations between the entries.
I'd like to prove that my matrix is always positive semidefinite and I'm just asking for some general pointers what I could look at. Some simple ideas that could work, perhaps. I tried to apply Gershgorin's circle theorem, but that doesn't always work. I read that this type of matrix is called an $L$ matrix but didn't find much information otherwise, maybe there is some useful theory about them around?