10
$\begingroup$

Let $S_n$ be the space of real $n \times n$ symmetric matrices and let $S_n^+$ be the convex cone of positive semidefinite matrices in $S_n$. The extremal rays of this cone correspond to the positive semidefinite matrices of rank $1$.

My problem is that I have a subcone $C$ of $S_n^+$ which is given by (finitely many) inequalities, which are not all linear (quadratic for example). In $C$ I have another subcone C' given by generators which correspond to rank $1$ matrices. Since I would like to compare $C$ and C', it would be nice to have inequalities defining C'. So my question is if there are known methods to determine such inequalities.

1 Answers 1

0

Any closed convex set in a locally convex topological vector space over $\mathbb R$ is the intersection of some family of half-spaces. In particular, any convex cone in ${\mathbb R}^{n \times n}$ is defined by a collection of linear inequalities (not necessarily equations). Moreover, since the space is separable, a countable collection will do. I don't know what else you would expect to say about it.

  • 0
    Sorry, I guess I was not clear with my question. I hope it is clearer now.2011-09-08