Another kind of the positivity of matrices 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?
 A: If your question were posed in the real case there is an easy answer. Note that any polynomial $p(\cdot)$ of even degree $2d$ can be written as $p(z)=x^TMx$, where $M$ is a Hermitian matrix and $x=(1,z,z^2,\ldots,z^d)$.
In this language your question is how to characterize the cone of nonnegative polynomials (in terms of the matrix $M$): by a classical Theorem of Hilbert, real univariate polynomials are nonnegative if and only if they are sums of squares. That means that $M$ is psd.
Unfortunately, such nice characterization only holds in very few cases for multivariate polynomials (see http://www.msri.org/attachments/workshops/327/553_Lecture-notes_week1_Blekherman.pdf, Theorem 1.2). Moreover, in general, deciding nonnegativity of real polynomials is an NP-hard problem, and therefore one should not expect a closed form description of the cone of such matrices $M$.
Finally, even though I can't figure out a way to describe the complex case within the real one, I suspect there is no nice characterization of this cone.
