TITLE: An inequality on trace of product of two matrices
QUESTION [17 upvotes]: Suppose we have two $n \times n$ positive semidefinite matrices, $A$ and $B$, such that $\mbox{tr}(A), \mbox{tr}(B) \le 1$. 
Can we say anything about $\mbox{tr}(AB)$? Is $\mbox{tr}(AB) \le 1 $ too?

REPLY [4 votes]: An extension to @dineshdileep 's answer can also be found here, in which it shows that:
$$tr(A^TB)\le \sqrt{tr(A^TA)tr(B^TB)}$$