Nonhomogeneous Karcher equations with vector fields on positive definite matrices
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.
Publication Source (Journal or Book title)
European Journal of Mathematics
Lim, Y., Hiai, F., & Lawson, J. (2021). Nonhomogeneous Karcher equations with vector fields on positive definite matrices. European Journal of Mathematics https://doi.org/10.1007/s40879-021-00469-6