Write $A$ as a block matrix $\pmatrix{B & C\cr C^* & D\cr}$ and correspondingly $v = \pmatrix{c\cr w\cr}$. Then $\langle A v, v \rangle = \langle B c, c \rangle + 2 \text{Re} \langle C^* c , w \rangle + \langle D w, w \rangle$ Now $\langle D w, w \rangle = \langle D^{1/2} w, D^{1/2} w \rangle$. If we write $w = D^{-1/2} u$ then $\langle A v, v \rangle = \langle Bc, c\rangle + 2 \text{Re} \langle D^{-1/2} C^* c, u \rangle + \langle u, u \rangle$ Now given $$, by Cauchy-Schwarz the minimum of $2 \text{Re} \langle D^{-1/2} C^* c, u\rangle$ with respect to $u$ is when $u$ is a negative multiple of $D^{-1/2} C^* c$. For $u = -t D^{-1/2} C^* c$ with $t > 0$ we have $\langle D^{-1/2} C^* c, u \rangle = - t \|D^{-1/2} C^* c \|^2$ and thus $ \langle A v, v \rangle = \langle Bc, c\rangle + (t^2 - 2 t) \|D^{-1/2} C^* c\|^2 $ To minimize $t^2 - 2 t$ we take $t = 1$, so that the minimum value of $\langle A v, v \rangle$ is $\langle B c, c \rangle - \|D^{-1/2} C^* c \|^2 = \langle (B - C D^{-1} C^*) c, c \rangle $