Solving symmetric matrix word equations via symmetric space machinery

It has recently been proved by Hillar and Johnson [C.J. Hillar, C.R. Johnson, Symmetric word equations in two positive definite letters, Proc. Amer. Math. Soc. 132 (2004) 945–953] that every symmetric word equation in positive definite matrices has a positive definite solution (existence theorem). In this paper we partially solve the uniqueness conjecture via the symmetric space and non-positive curvature machinery existing in the open convex cone of positive definite matrices. Unique positive definite solutions are obtained in terms of geometric and weighted means and as fixed points of explicit strict contractions on the cone of positive definite matrices.