Quadratic/Bilinear form positive definiteness

Question

I've the following interesting (or that's what I think at least!) question. Let $B\in\mathbb{R}^{p\times n}$, $A\in\mathbb{R}^{n\times n}$ and $C\in\mathbb{R}^{n\times p}$. I'm interested in the sign of following operation $$ x^{T} BAC x$$ with $x\in\mathbb{R}^{p\times 1}$. Further assumptions:

• The symmetric part of $A$ is positive definite, i.e. $(A+A^T)/2 > 0$
• $B$ is full (row) rank
• $C$ is full (column) rank

Edit: I've changed the question as per the useful first comment.

Here comes the question: Is there any relatively "straightforward" additional condition on $B$ and $C$ to guarantee that $x^{T}BACx > 0$ under the hypotheses stated above? (of course I'm not interested in $B=C$).

Thanks in advance!!

Answer 1

No. Suppose $x^{T}BACx$ is a positive definite quadratic form. Then $x^{T}BA(-C)x$ is a negative definite quadratic form and $-C$ satisfies the same requirements as $C$.