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We show that, for each value of α∈(-1, 1), the only Riemannian metrics on the space of positive definite matrices for which the ∇(α) and ∇(-α) 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 Wigner–Yanase–Dyson metric.

DUALITY, MONOTONICITY AND THE WIGNER–YANASE–DYSON METRICS