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!

In fact, you can show that $\mathrm{Re}(z^*Mz) > 0$ for every non-zero complex vector $z$. Indeed, suppose that you can find a $z$ for which it doesn't hold:
$$\mathrm{Re}(z^*Mz) \leq 0.$$
Then you have
\begin{align*}
\mathrm{Re}(z^*M^*z) &= \mathrm{Re}((z^*Mz)^*)\\
&= \mathrm{Re}(z^*Mz)\\
&\leq 0.
\end{align*}
However, you then get
$$\mathrm{Re}(z^*(M + M^*)z) \leq 0, $$
which contradicts the positive-definiteness of the Hermitian part.
