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