ar X iv : m at h / 02 09 39 9 v 1 [ m at h . R A ] 2 9 Se p 20 02 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 fixed 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 define 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.