# positive semi definite matrix problem

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?

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For part (1), consider the fact that $A$ being positive definite means $$\mathbf{x}^\mathrm{T}A\mathbf{x} > 0$$ for all $\mathbf{x}$. Can you conclude something similar with $B^\mathrm{T}AB$?
(1) I think what you mean is that $B^TAB$ is positive semi-definite, which is easy to show if notice that $$B^TAB=B^TA^{1/2}A^{1/2}B=(A^{1/2}B)^TA^{1/2}B,$$ where $A^{1/2}$ is the square root of $A$, so ${A^{1/2}}^T=A^{1/2}$.