# Positive definite matrix properties

I am having trouble solving a property that I found.

If $$A:n \times n$$ is defined as a positive definite matrix and $$B: n \times m$$ where $$rank(B) = r$$.

Then $$B^T A B > 0$$, only when r = m and $$B^T A B \ge 0$$, when $$r < m$$.

$$B^TAB>0$$ iff $$(Bx)^TA(Bx)=x^TB^TABx>0$$ for all nonzero vectors $$x\in\mathbb R^m$$. Since $$A$$ is positive definite, the question now becomes whether $$Bx$$ could be zero for some nonzero vector $$x$$. This is where the rank of $$B$$ matters.