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.