Is $AB+BA$ positive definite too if $A$ and $B$ are positive definite?
I have a question: Is $AB+BA$ positive definite too if $A$ and $B$ are positive definite matrices?
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 $-1$.