TITLE: Does the product of positive definite matrices have a positive trace
QUESTION [7 upvotes]: Let $ A_{1}, A_{2}, \ldots, A_{m}$ be a real symmetric semi-positive  definite matrices,  I want to know whether 
$ tr (A_{1} \cdot A_{2} \cdots A_{m} ) \geq 0$ ?
When $m=2$, it seems a rather standard problem and has a "yes" answer.

REPLY [6 votes]: For $m\ge3$, no. Counterexample:
$$
A = \pmatrix{6&-2\\ -2&1},
\ B = \pmatrix{1&-2\\ -2&8},
\ C = \pmatrix{10&8\\ 8&10},
\ ABC = \pmatrix{-124&-200\\ 56&88}.
$$
Note that $A,B,C$ are positive definite.