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.

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$$
{\rm Tr}\pars{AB}
=
\sum_{ij}A_{ij}B_{ji} < \sum_{ij}A_{ij}\quad\mbox{if}\quad A_{ij} > 0, \forall i,j.\
\mbox{So, what ?}
$$
