Let $A$ be a symmetric positive matrix, and let $B$ be invertible. Is
$$BAB^{1} + B^{1}AB$$
always positive?
Let $C$ be a real matrix with real positive spectrum. Is
$$C + C^T$$
positive?
Are these two problems the same?
Let $A$ be a symmetric positive matrix, and let $B$ be invertible. Is $$BAB^{1} + B^{1}AB$$ always positive? Let $C$ be a real matrix with real positive spectrum. Is $$C + C^T$$ positive? Are these two problems the same? 


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. 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. 


2nd question: obviously NO. Just take $$C=\begin{pmatrix} 1 & 0 \\\\ a & 1 \end{pmatrix}$$ whose spectrum is $(1)$. If $a>2$, then $C+C^T$ is not positive. The questions are not the same, because on the one hand, $(BAB^{1})^T$ is not equal, in general, to $B^{1}AB$. 

