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