Z-transformations on proper and symmetric cones

Motivated by the similarities between the properties of Z-matrices on $R^{n}_+$ and Lyapunov and Stein transformations on the semidefinite cone $mathcal {S}^n_+$ , 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.