Positive Definite Matrix Multiplication Non-Symmetric

Question

If we have $B$ is an $(n,n+1)$ matrix and $A$ is a positive definite $(n,n)$ matrix is the resulting quantity positive definite?

$$B^TAB$$

It seems like it should be since we have both a $B^T$ and a $B$ in there and we know that $B^TB$ is positive definite, but I am getting stuck on an explicit proof.

Thanks!

Answer 1

It is never positive definite. $B$ is a fat matrix. Thus $Bx=0$ has a nontrivial solution $x$ and $x^TB^TABx=0$.