Symmetric word equations in two positive definite letters

For every symmetric (``palindromic") word $S(A,B)$ in two positive definite letters and for each fixed $n$-by-$n$ positive definite $B$ and $P$, it is shown that the symmetric word equation $S(A,B) = P$ has an $n$-by-$n$ positive definite solution $A$. Moreover, if $B$ and $P$ are real, there is a real solution $A$. The notion of symmetric word is generalized to allow non-integer exponents, with certain limitations. In some cases, the solution $A$ is unique, but, in general, uniqueness is an open question. Applications and methods for finding solutions are also discussed.