Does it follow that the hermetian part of a matrix is positive definite, that the matrix itself is invertible? I came across this in a paper and I was wondering whether it is true. We have a complex matrix $M$ such that $(M+M^H)$ is positive definite. Now, it is clear that $(M+M^H)$ is invertible, but does that hold for $M$?
Is there (in general) some connections between the hermetian part of a matrix and the hermetian part of its inverse?
Thanks guys!

\begin{eqnarray}
\langle x , (M+M^*) x \rangle &=& \langle x , M x \rangle + \langle x , M^* x \rangle  \\
&=&  \langle x , M x \rangle + \overline{\langle M^* x , x \rangle}  \\
&=& \langle x , M x \rangle + \overline{\langle  x , M x \rangle}  \\
&=& 2 \operatorname{re} \langle x , M x \rangle
\end{eqnarray}
In particular, $\langle x , M x \rangle \neq 0 $ for all $x \neq 0$.
