If $A$ Hurwitz, $(A+A^*)$ is Hurwitz?

Question

If I have $A$ Hurwitz matrix, is $(A+A^*)$, with $A^*$ the transpose of $A$, still Hurwitz? Any reference or proof?

Because if $(A+A^*)$ is still Hurwitz I can say that it is even negative definite being symmetric and with real eigenvalues negative.

Answer 1

No, the sum with adjoint (or, equivalently, the symmetric/Hermitian part) of a Hurwitz matrix is not in general a Hurwitz matrix. An example was given by Daniel Fischer in comments, and it can look like this: $$\begin{pmatrix} -1 & 10 \\ 0 & -1 \end{pmatrix}$$ is Hurwitz (both eigenvalues are $-1$) but its sum with transpose is $$\begin{pmatrix} -2 & 10 \\ 10 & -2 \end{pmatrix}$$ which is a symmetric matrix with negative determinant. Such a matrix has real eigenvalues of opposite signs.