For given $n$, the following $n\times n$ complex matrix $M=M^{\dagger}$ is called *positive*, if

$x^{\dagger}M x\geq 0$

holds for all complex vector $x=(z,z^2,\cdots,z^n)^T$ with arbitrary complex $z$, where $M^{\dagger}$ stands for the conjugate transpose of $M$.

Please note that this definition of positivity is not the same as positive definiteness since $x$ is nor ranging over all vectors here.

Of course, the set of such positive matrices forms a convex cone, I would like to have a closed form of the cone, for a given $N$, how to check whether $N$ is positive?

eachlinear algebra textbook. you can look for answers there. $\endgroup$ – Delio Mugnolo Sep 13 '14 at 19:50say the conjugate transpose$\endgroup$ – Yemon Choi Sep 14 '14 at 12:00