#### Title

Nonhomogeneous Karcher equations with vector fields on positive definite matrices

#### Document Type

Article

#### Publication Date

1-1-2021

#### Abstract

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

#### Recommended Citation

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