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 RnĂ—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.
can we find general form for elements of intersection of positive semidefinite matrices with convex cones of other matrices?
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convex-analysis