Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices
Bryan E. Cain, Daniel Hershkowitz, Hans Schneider · Czechoslovak Mathematical Journal · 1997
Standard facts about separating linear functionals will be used to determine how two cones C and D and their duals C * and D * may overlap. When T : V → W is linear and K ⊂ V and D ⊂ W are cones, these results will be applied to C = T ( K ) and D , giving a unified treatment of several theorems of the alternate which explain when C contains an interior point of D . The case when V = W is the space H of n × n Hermitian matrices, D is the n × n positive semidefinite matrices, and T ( X ) = AX + X* A yields new and known results about the existence of block diagonal X 's satisfying the Lyapunov condition: T ( X ) is an interior point of D. For the same V , W and D , T ( X ) = X − B * XB will be studied for certain cones K of entry-wise nonnegative X 's.