Conditions for a (2x2) block matrix to be Positive definite Let us consider a given symmetric matrix
\begin{equation}
M=\begin{bmatrix}0 & B\\
B^{T} & C
\end{bmatrix}.
\end{equation}
My question may seem very simple but what conditions must the block
matrices follow in order to make the big matrix positive definite
please? A, B and C are matrices. 
Thanks. 
 A: I'm creating an answer from the comments.

If $M$ has a zero subblock in the (1,1) position then $M$ cannot be positive definite because
$$ \begin{bmatrix} v^T & 0 \end{bmatrix}M\begin{bmatrix} v \\ 0 \end{bmatrix} = v^T0v = 0. $$
Nor can it have a zero subblock in the (2,2) position because then it is conjugate to a matrix with a zero subblock in the (1,1) position:
$$ \begin{bmatrix} 0 & I \\ I & 0 \end{bmatrix} \begin{bmatrix}A & B\\ B^{T} & 0 \end{bmatrix} \begin{bmatrix} 0 & I \\ I & 0 \end{bmatrix}^{-1} = \begin{bmatrix} 0 & B^T \\ B & A \end{bmatrix}. $$
You can also see this directly
$$ \begin{bmatrix} 0 & v^T \end{bmatrix} \begin{bmatrix} A & B\\ B^{T} & 0 \end{bmatrix} \begin{bmatrix} 0 \\ v \end{bmatrix} = v^T0v = 0. $$

If the (1,1) subblock is $0$ and $M$ is positive semidefinite then the $(1,2)$ and $(2,1)$ must be zero as a consequence of this result. This leaves us with
$$ \begin{bmatrix}0 & 0\\0 & C\end{bmatrix} $$
which is positive semidefinite iff $C$ is positive semidefinite.
