Prove $A^TB$ is a positive semi-definite matrix?

Question

$A,B \in R^{m\times n}$ and the singular values of both $A$ and $B$ are between 0 and 1. Is $A^TB$ a positive semi-definite matrix? Please show me the proof:)

Answer 1

Of course not. You can always rescale $A$ and $B$ so their singular values are between $0$ and $1$, but that doesn't affect positive semidefiniteness of $A^T B$.

Answer 2

Try $$A=1, \qquad B=-1$$ $1\times 1$ matrices