I hope this question is sufficiently interesting for you to try to answer it. Sorry if its too trivial, but, working with a few similarty transformations I've arrived to the following conclusion, and I want to know if its true.
Consider that $A$ is a companion matrix. The symmetric-part of $A$ can never be either positive-definite or negative-definite, by using Sylvester's law of inertia, since at least the first minor of the symmetric-part will be always zero. Now the interesting part: if I don't remember bad, a matrix $B$ is similar to a companion form if and only if all its eigenvalues have geometric multiplicy one (i.e. there is only one Jordan block for each eigenvalue in the Jordan decomposition of $B$). So.. the question is... if a matrix $B$ is similar to a companion form, its symmetric part will always have eigenvalues in both sides of the complex plane?
That's not great news for me, at all, since I've a matrix of these caracteristics within an optimization process. I'm trying to find conditions for the convexity of my problem, and now I came up with this.
Thanks in advance!!!!