  Farkas' lemma for semidefinite programming characterizes semidefinite
feasibility of linear matrix pencils in terms of an alternative spectrahedron.
In the well-studied special case of linear programming, a theorem by Gleeson
and Ryan states that the index sets of irreducible infeasible subsystems are
exactly the supports of the vertices of the corresponding alternative
polyhedron.
  We show that one direction of this theorem can be generalized to the
nonlinear situation of extreme points of general spectrahedra. The reverse
direction, however, is not true in general, which we show by means of
counterexamples. On the positive side, an irreducible infeasible block
subsystem is obtained whenever the extreme point has minimal block support.
Motivated by results from sparse recovery, we provide a criterion for the
uniqueness of solutions of semidefinite block systems.
