Symmetric Word Equations in Two Positive Definite Letters
  A generalized word in two positive definite matrices A and B is a finite product of nonzero real powers of A and B. Symmetric words in positive definite A and B are positive definite, and so for fxed B, we can view a symmetric word, S(A,B), as a map from the set of positive definite matrices into itself. Given positive definite P, B, and a symmetric word, S(A,B), with positive powers of A, we defne a symmetric word equation as an equation of the form S(A,B) = P. Such an equation is solvable if there is always a positive definite solution A for any given B and P. We prove that all symmetric word equations are solvable. Applications of this fact, methods for solution, questions about unique solvability (injectivity), and generalizations are also discussed. 
Christopher J. Hillar and Charles R. Johnson