Positive and completely positive cones and Z-transformations
نویسندگان
چکیده
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 is also positive definite. A recent result of Moldovan-Gowda says that a Z-matrix A is positive stable if and only if for some symmetric strictly copositive matrix X, AX + XA is also strictly copositive. In this paper, these results are unified/extended by replacing R and R + by a closed convex cone C satisfying C−C = R. This is achieved by relating the Z-property of a matrix on this cone with the Z-property of the corresponding Lyapunov transformation LA(X) := AX+XA T on the completely positive cone of C and the Z-property of L AT on the copositive cone of C in S (the space of all real n × n symmetric matrices). A similar analysis is carried out for the Stein transformation SA(X) = X − AXA T .
منابع مشابه
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