Symmetric Γ-submanifolds of positive definite matrices and the Sylvester equation XM=NX

In this paper we consider the special Sylvester equation XM-NX=0 for fixed n×n matrices M and N, where a positive definite solution X is sought. We show that the solution sets varying over (M,N) provide a new family of geodesic submanifolds in the symmetric Riemannian manifold Pn of positive definite matrices which is stable under congruence transformations; it consists of geodesically complete convex cones of Pn invariant under Cartan symmetries. It is further shown that the solution set is stable under the iterative means obtained by the weighted arithmetic, harmonic and geometric means.