Z-transformations on proper and symmetric cones (Q959969)

The authors introduce and study \(\mathbf{Z}\)-transformations on proper cones. They show that many properties of \(\mathbf{Z}\)-matrices extend to \(\mathbf{Z}\)-transformations. It is shown that Lyapunov and Stein transformations have the \(\mathbf{Z}\)-property on the semidefinite cone, and that many of the properties of \(\mathbf{Z}\)-matrices generalize to proper cones. In particular, the authors focus on the complementarity properties. They describe the diagonal stability of a \(\mathbf{Z}\)-transformation on a symmetric cone by means of quadratic representations. They also study the equivalence of \(\mathbf{Q}\) and \(\mathbf{P}\) properties of \(\mathbf{Z}\)-transformations on symmetric cones. In particular, they prove such an equivalence on the Lorentz cone.