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On the solution of the nonlinear matrix equation \(X^n=f(X)\). (English) Zbl 1165.15015

Authors’ summary: We consider a class of nonlinear matrix equations \(X^n-f(X)=0\) where \(f\) is a self-map on the convex cone \(P(k)\) of \(k\times k\) positive definite real matrices. It is shown that for \(n\geqslant 2\), 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\((k,\mathbb R)\)-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 \(n\geqslant 2\) converges always to the identity matrix.

MSC:

15A24 Matrix equations and identities
15A23 Factorization of matrices
15B48 Positive matrices and their generalizations; cones of matrices
65F30 Other matrix algorithms (MSC2010)
65H10 Numerical computation of solutions to systems of equations
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.

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