Let $A$ be a symmetric positive matrix, and let $B$ be invertible. Is


$$BAB^{-1} + B^{-1}AB$$


always positive?




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Let $C$ be a real matrix with real positive spectrum. Is


$$C + C^T$$


positive?




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Are these two problems the same?



Positivity and symmetrization
*EDIT:* I changed the counterexample to ensure that $B$ is also *symmetric*


**First question**: NO, not even if $B$ is also (symmetric) positive definite


Take


$$ A = \begin{pmatrix}
5 & 2\\\\
2 & 4\end{pmatrix}\qquad
B = \begin{pmatrix}
13 & 15\\\\
15 & 18\end{pmatrix}
$$


Here, both $A$ and $B$ are (symmetric) positive definite, but the sum
$$BAB^{-1} + B^{-1}AB = \begin{pmatrix}
93.3 & 17.9\\\\
17.9 & -75.3\end{pmatrix},
$$
which is not positive.




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**Added** You might want to generalize the following weaker statement: If $A$ and $B$ are Hermitian, $A$ is positive, and $AB+BA$ is positive, then the matrix $B$ is also positive definite.


