TITLE: Product of positive-definite matrices has positive trace
QUESTION [9 upvotes]: In his paper "On the existence of a connection with curvature zero", Milnor makes the following claim.  Let $X$ and $Y$ be positive-definite $n \times n$ matrices.  Then the trace of $X Y$ is positive.  I can't figure out how to prove this.  Can someone help me?

REPLY [18 votes]: Let $A$ be the positive definite square root of $X$ and $B$ the positive definite square root of $Y$.
You have
$$
\mbox{tr}(XY)=\mbox{tr}(AABB)=\mbox{tr}(BAAB)=\mbox{tr}((AB)^*AB)>0.
$$
Indeed, the latter is the sum of all $c_{i,j}^2$ where $c_{i,j}=(AB)_{i,j}$.
So it is nonnegative. 
And if it were zero, this would imply $AB=0$ hence $A=B=0$ since they are both invertible. Contradiction.