TITLE: Is the product of two positive semidefinite matrices positive semidefinite?
QUESTION [6 upvotes]: If $X$ and $W$ are real, square, symmetric, positive semidefinite matrices of the same dimension, does $XW + WX$ have to be positive semidefinite?
This is not homework.

REPLY [7 votes]: If $A, B$ are real, pos, and symmetric, then $A=A^{1/2}A^{1/2}$ and the trace of $AB$ is the trace of $A^{1/2}A^{1/2}B$ which is the the trace of $A^{1/2}BA^{1/2}$ which is a positive semidefinite matrix. Thus trace of $AB$ is nonnegative.