TITLE: Is $AB+BA$ positive definite too if $A$ and $B$ are positive definite?
QUESTION [12 upvotes]: I have a question: Is $AB+BA$ positive definite too if $A$ and $B$ are positive definite matrices?

REPLY [22 votes]: It is false in general. For example, let $A=\begin{pmatrix}2 &1\\1&1\end{pmatrix}$ and $B=\begin{pmatrix}6 &0\\0&1\end{pmatrix}$. Then 
$$AB+BA=\begin{pmatrix}24 &7\\7&2\end{pmatrix},$$
whose determinant is $-1$.