  A generalized word in two letters $A$ and $B$ is an expression of the form
$W=A^{\alpha_1}B^{\beta_1}A^{\alpha_2}B^{\beta_2}... A^{\alpha_N}B^{\beta_N}$
in which the exponents $\alpha_i$, $\beta_i$ are nonzero real numbers. When
independent positive definite matrices are substituted for $A$ and $B$, we are
interested in whether $W$ necessarily has positive eigenvalues. This is known
to be the case when N=1 and has been studied in case all exponents are positive
by two of the authors. When the exponent signs are mixed, however, the
situation is quite different (even for 2-by-2 matrices), and this is the focus
of the present work.
