Let $ A \in \mathbb{R}^{n \times m}, $ with $1<m<n$, where $ \text{rank}(A)=m $ (i.e., $ A $ is a tall matrix of full column rank). Let $ B \in \mathbb{R}^{n \times n} $ be a positive definite matrix. Can we say anything about the definiteness of the product $ A^{\text{T}}BA $?
My gut feeling is since $A$ is of full column rank and $m<n$ that $ A^{\text{T}}BA $ is positive definite; however, I cannot find a way to show this. I have seen many posts about the definiteness of matrix product where all matrices are either positive definite or positive semi definite; however, I cannot seem to find anything quite like the question I have presented.
A little background, I am working on some research where I need $ A^{\text{T}}BA $ to be invertible. If invertibility cannot be guaranteed then I need to take another approach.