Nonhomogeneous Karcher equations with vector fields on positive definite matrices

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We study a family of Riemannian gradient equations on the Cartan–Hadamard–Riemannian manifold PN of N×N positive definite Hermitian matrices ∇Rie[12∑k=1nδ2(X,Ak)]=F(X),where δ(A, B) denotes the Riemannian distance between A and B and F varies over differentiable vector fields on PN. Our equations give rise to a number of nonlinear matrix equations. The special case where F(X) = 0 is the vanishing gradient equation (called the Karcher equation) of the sum of the squares of the distances, whose unique solution is the Karcher mean of A1, … , An. In particular, when n= 1 , the equation is closely related to the matrix Lambert W function. A class of vector fields for which the equation admits a (unique) solution is presented, including the constant vector fields, the vector fields of positive congruence transformations, and those given in terms of the gradients for several kinds of functions.

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European Journal of Mathematics

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