Given
$$ C = AFA^T + A\bar{F} $$
where $A = [A_1 A_2]$, $F = \begin{bmatrix} F_1 & F_2 \\ F_2^T & F_3 \end{bmatrix}$, $\bar{F} = 2 \begin{bmatrix} \bar{F}_1 \\ \bar{F}_2 \end{bmatrix}$ such that $A_i \in \mathbb{R}^{n \times k}, F_i = F_i^T \in \mathbb{R}^{k \times k}, \bar{F}_i \in \mathbb{R}^{k \times n}$ and further $F_i, C = C^T \succeq 0$ with $k \leq n$.
Is it possible to represent $C \succeq 0$ in the following compact-quadratic form:
$$ C = A_M F_M A_M^T $$
for some $A_M$ with $F_M = F_M^T \succeq 0$?