Suppose one is given matrices $A_1, A_2, \dots, A_w \in \mathbb{R}^{r \times n}$ and matrix $B \in \mathbb{R}^{n \times n}$. When and how can one construct matrix $C \in \mathbb{R}^{r \times n}$ such that $ABC$ is positive definite, i.e., the following holds?
$$\lambda_{\min} \Big ((\frac{1}{w} \sum_{i=1}^w A_i)BC^\top \Big ) > 0$$
If (a) $r \leq n$, (b) $\frac{1}{w} \sum_{i=1}^w A_i$ is full-rank i.e rank $r$ and (c) $B$ is PD then a choice is $$C = \frac{1}{w} \sum_{i=1}^w A_i$$ Are there any weaker conditions when a choice of $C$ is obvious?
Is any condition getting implicitly enforced between $w,r$ and $n$ to get this product to be PD?
As soon as the rank of matrix $X=\frac1w\sum_{i=1}^wA_iB$ is r, we could choose $C=X$