TITLE: Does a positive semidefinite matrix always have a non-negative trace?
QUESTION [16 upvotes]: If $A$ is a positive semidefinite matrix ($A\succeq 0)$, does it imply that $\mbox{Tr}(A)\geq 0$, where the $\mbox{Tr}(\cdot)$ denotes the trace.

If not, any counter-example? Thanks.

REPLY [15 votes]: Yes. If the matrix is semi-positive definite, all the eigenvalues are non-negative. The trace being equal to the sum of eigenvalues, we conclude that the trace has to be non-negative.