Invariant metrics, contractions and nonlinear matrix equations

In this paper we consider the semigroup generated by the self-maps on the open convex cone of positive definite matrices of translations, congruence transformations and matrix inversion that includes symplectic Hamiltonians and show that every member of the semigroup contracts any invariant metric distance inherited from a symmetric gauge function. This extends the results of Bougerol for the Riemannian metric and of Liverani–Wojtkowski for the Thompson part metric. A uniform upper bound of the Lipschitz contraction constant for a member of the semigroup is given in terms of the minimum eigenvalues of its determining matrices. We apply this result to a variety of nonlinear equations including Stein and Riccati equations for uniqueness and existence of positive definite solutions and find a new convergence analysis of iterative algorithms for the positive definite solution depending only on the least contraction coefficient for the invariant metric from the spectral norm.