  We show that, for each alpha in the interval (-1,1), the only Riemannian
metrics on the space of positive definite matrices for which the alpha and
-alpha-connections are mutually dual are matrix multiples fo the
Wigner-Yanase-Dyson metric. If we further impose that the metric be monotone,
then this set is reduced to scalar multiples of the Wigner-Yanase-Dyson metric.
