2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/127571In this paper we develop the theory of information geometry for single random matrix models, with two goals: proving a Cramer-Rao theorem for estimators on random matrices, and calculating the Legendre transform of pressure and entropy with respect to a metric duality. Consequently, in the large n limit we recover several quantities from free probability: Voiculescu's conjugate variable is the tangent vector to the GUE perturbation model, giving rise to a metric which turns out to be the free Fisher information measure; Hiai's Legendre transform of free pressure agrees with our Legendre transform of pressure; and Speicher's covariance of fluctuations naturally arises as the metric on the random matrix model obtained from the fluctuation functions.29 pages, no figures; corrected typos, a few sections revisedOperator AlgebrasProbability15A52 Random matricestext