Let $A \in \mathbb R^{n\times n}$ and assume that all eigenvalues lie in the left open halfplane. 
Is it true that the logarithmic norm $\mu\_2 (A):= \lambda\_{\max}
\left(\frac{A + A^T}{2}\right)<0?$



Does stability imply that the logarithmic norm is negative?
No. Take
$$A=\left(\begin{array}{cc}a&b\\ 0&a\end{array}\right),$$
where $a$ is negative but $b$ is positive and large.
The largest eigenvalue of $(A^T+A)/2$
is $a/2+b/2$.


