... stands for the spectral norm. The Thompson metric exists on any opennormal convex cones of r eal Banach spaces [5,6]; in particular, the open convex cone of positive definite operat ors of a Hilbert ... son [5] for thestudy of solutions to systems of nonlinear matrix equations involving contractivemappings.We first review the Thompson metric on the open convex cone P(n)(n ≥ 2), the set of all ... invariant properties of the metric, we then haved(MXrM∗, MYrM∗)≤|r|d(X, Y), r ∈ [−1, 1](4) for any X, Y Î P(n) and nonsingular matrix M .Lemma 1.1 (see [8]). For all A, B, C, D...