On sep 8, 2011 a question was asked about cones of positive semidefinite matrices that can be generated by rank 1 matrices. A respondent answered "any convex cone in Rn×n is defined by a collection of linear inequalities (not necessarily equations). Moreover, since the space is separable, a countable collection will do". I have a similar question. I need, if possible, to find a general form for elements of the intersection of positive semidefiite matrices with other convex cones of matrices. Is it true that the general form is a conic combination of certain extremal or rank one matrices, and if so, is it possible to apply Caratheodory Theorem? If not, what other general form is possible.