Motivated by the similarities between the properties of Z-matrices on the nonnegative orthant and Lyapunov and Stein transformations on the semidefinite cone, in this article, we introduce and study Z-transformations on proper cones. We show that many properties of Z-matrices extend to Z-transformations. We describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations. Finally, we study the equivalence of Q and P properties of Z-transformations on symmetric cones. In particular, we prove such an equivalence on the Lorentz cone.
(January 31, 2006, revised December 18, 2006)