Trace of matrices inequality If I have two matrices, $\mathbf{A}$ which is symmetric and postive definite, and $\mathbf{B}$ symmetric, positive definite, and all entries in $\mathbf{B}$ are between 0 and 1, with the diagonal entries all equal to 1.
I can't seem to see why tr($\mathbf{AB}$) $\leq$ tr($\mathbf{A}$).
Thanks.

Not true.  Try $A = B = \pmatrix{1 & t\cr t & 1\cr}$ for $0 < t < 1$.
