On positive and completely positive cones and Z-transformations

Abstract

A well-known result of Lyapunov on continuous linear systems asserts that a real square matrix A is positive stable if and only if for some symmetric positive definite matrix X, AX+XA^T is also positive definite. A recent result of Moldovan-Gowdasays that a Z-matrix A ispositive stable if and only if for some symmetric strictly copositive matrix X, AX+XA^T is also strictly copositive. In this paper, these results are unified/extended by replacing Rⁿ and Rⁿ⁺ by a closed convex cone C satisfying C−C=Rⁿ. This is achieved by relating the Z-property of a matrix on this cone with the Z-property of the corresponding Lyapunov transformation L_A(X) := AX+XA^T on the completely positive cone of C and the Z-property of L_A^T on the copositive cone of C in Sⁿ (the space of all real n×n symmetric matrices).  A similar analysis is carried out for the Stein transformation S_A(X) =X−AXA^T.