Is there a way to generate a random matrix that satisfies a given linear matrix inequality (LMI)?

The title is self-evident, I think. For example, we could generate a $n \times n$ random rank-$1$ positive semidefinite matrix by generating a random vector $x$ and $x x^T$ will be the random matrix we want. Now if there is a given linear matrix inequality, i.e., a spectrahedron, is there a way to generate a matrix that is included in the spectrahedron?

So to make it more clear. Consider symmetric matrix space $S^n$,we can generate a random positive definite matrix $X$ in this space by, for example, generate some matrix $A$ and let $X=A^TA$,but what if I want to generate a random positive definite matrix in a certain subspace of the matrix space, such as defined by a Linear Matrix Inequality (as Omnomnomnom suggested in the comments). Is there a way to do it?

Thank you!

• This could be made clearer. Commented Jul 5, 2015 at 16:05
• I suppose you're specifically referring to linear matrix inequalities Commented Jul 5, 2015 at 16:06