G. R. Duan, 1998 shows that the set $\mathcal{S}_n$ of real $n \times n$ Hurwitz-stable matrices (defined as those having eigenvalues whose real part are strictly negative) is a simply-connected, convex cone in $\mathbb{R}^{n \times n}$ with vertex at the origin.
My question is this: Can we find the exact location of the boundary of $\mathcal{S}_n,$ given some specific order of the axes (e.g., the first $n$ axes correspond to the value of the $n$ entries in the first row of a given matrix, the next $n$ axes correspond to the next row, etc.)?
If it is not possible to give an explicit parameterization of the boundary (or a list of inequalities defining it), is it then possible to determine other geometrical features of it, like
- whether or not the cone is a right cone (it has zero eccentricity),
- if its axis coincides with one of the axes of $\mathbb{R}^{n \times n}$ or if it is off to an angle,
- what its "opening angle" is, etc.?
I apologize if these questions are either trivial or ill-defined. I'm not formally trained in mathematics, so I may have been relying too much on my geometrical intuition when formulating them. I don't believe $\mathcal{S}_n$ is finitely generated, but instead looks something like this (here in the 3D analogue):
Any ideas and/or references would be of great help!
My alternative to finding the exact parameterization of the boundary would be to simulate it for a specific dimension $n$: I'd take a matrix within $\mathcal{S}_n$ and vary the entries (almost) continuously until I hit the boundary (when my matrix is no longer Hurwitz-stable). Doing this a lot of times, I'd get some points that are on the boundary, but I would then have to fit that to some sort of $n^2$-dimensional cone, which I don't know much about, and that seems like a daunting task. Hence this question.
Thank you for your time!