Theorems of the Alternative for Cones and Lyapunov Regularity of Matrices


    loading  Checking for direct PDF access through Ovid

Abstract

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: VW is linear and KV and DW 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) = XB* XB will be studied for certain cones K of entry-wise nonnegative X's.

    loading  Loading Related Articles