On the solution of the nonlinear matrix equation Xn=f(X)

We consider a class of nonlinear matrix equations Xn-f(X)=0 where f is a self-map on the convex cone P(k) of k×k positive definite real matrices. It is shown that for n⩾2, the matrix equation has a unique positive definite solution depending continuously on the function f if f belongs to the semigroup of nonexpansive mappings with respect to the GL(k,R)-invariant Riemannian metric distance on P(k), which contains congruence transformations, translations, the matrix inversion and in particular symplectic Hamiltonians appearing in Kalman filtering. We show that the sequence of positive definite solutions varying over n⩾2 converges always to the identity matrix.