$M$ is congruent to ${\rm diag}(A-BC^{-1}B^T,C)$. Therefore the condition $A-BC^{-1}BT\le0$ amounts to saying that the maximal dimension of positive subspace is the size $p$ of $C$.If $A-BC^{-1}BT<0$, this is saying that the signature of $M$ is $(p,0,q)$, where $q$ is the size of $A$.
Denis Serre
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