Abstract
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 P n of positive definite matrices which is stable under congruence transformations; it consists of geodesically complete convex cones of P n 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.
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