Claim: $\nabla^2 (x^TAx + 2x^TBy + y^TCy) = \begin{bmatrix} A & B \\ B^T & C \end{bmatrix}$ https://en.wikipedia.org/wiki/Schur_complement#Schur_complement_condition_for_positive_definiteness_and_positive_semidefiniteness https://inst.eecs.berkeley.edu/~ee127a/book/login/thm_schur_compl.html
I cannot seem to get the correct formas above...
Here is the corresponding calculation
$f(x,y) = x^TAx + 2x^TBy + y^TCy$
$\dfrac{\partial f}{\partial x_k} = 2 \sum\limits_{i = 1}^n x_i A_{ik} + \sum\limits_{i = 1}^n 2 y_j B_{kj}$
$\dfrac{\partial^2 f}{\partial x_l \partial x_k} = 2 A_{lk}$
$\dfrac{\partial f}{\partial y_k} = 2 \sum\limits_{i = 1}^n x_i B_{ik} + \sum\limits_{i = 1}^n 2 y_i C_{ik}$
$\dfrac{\partial^2 f}{\partial y_l \partial y_k} = 2 C_{lk}$
$\dfrac{\partial^2 f}{\partial y_l \partial x_k} = 2 B_{kl}$
$\dfrac{\partial^2 f}{\partial x_l \partial y_k} = 2 B_{lk}$
$\nabla^2 f(x,y) = \begin{bmatrix} \dfrac{\partial^2 f}{\partial x_l \partial x_k} & \dfrac{\partial^2 f}{\partial y_l \partial x_k} \\ \dfrac{\partial^2 f}{\partial x_l \partial y_k} & \dfrac{\partial^2 f}{\partial y_l \partial y_k} \end{bmatrix} = \begin{bmatrix} 2A & 2B^T \\ 2B & 2C \end{bmatrix}$