Let $A$ be a $p\times p$ positive semi definite matrix.
(1) If $A$ is positive definite, and $B$ is a $p\times q$ matrix with rank $p\leq q$. Show that $B^TAB$ is positive definite.
(2) If $rankA=s\leq p$. Show that there exists a $p\times s$ matrix $M$ such that $A=MM^T$ and $M^TM$ is a diagonal matrix of the positive eigenvalues of A.
I don't even know how to start. Can someone give some hints?