Consider the word equation where are positive definite complex Hermitian -matrices, is the unknown matrix, and is a symmetric (“palindromic”) generalized word of the form ; here , , . “Symmetric” means that .
The authors show that every symmetric word equation is solvable and they conjecture uniqueness of the solution. In some cases the solution is unique (example: the unique solution of the equation is ); in general, uniqueness is an open question. If and are real, then one can find a real solution as well. The notion of symmetric word is generalized to allow non-integer exponents, with certain limitations. Applications and methods for finding solutions are also discussed.