Symmetric Word Equations in Two Positive Definite Letters

CHRISTOPHER J. HILLAR AND CHARLES R. JOHNSONAbstract. A generalized word in two positive definite matrices A and B isa finite product of nonzero real powers of A and B. Symmetric words inpositive definite A and B are positive definite, and so for fixed B, we canview a symmetric word, S(A,B), as a map from the set of positive definitematrices 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 asan equation of the form S(A,B) = P. Such an equation is solvable if there isalways a positive definite solution A for any given B and P. We prove that allsymmetric word equations are solvable. Applications of this fact, methods forsolution, questions about unique solvability (injectivity), and generalizationsare also discussed.