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We study a Riemannian metric on the cone of symmetric positive-definitematrices obtained from the Hessian of the power potential function$(1-\det(X)^\beta)/\beta$. We give explicit expressions for the geodesics anddistance function, under suitable conditions. In the scalar case, the geodesicbetween two positive numbers coincides with a weighted power mean, while formatrices of size at least two it yields a notion of weighted power meandifferent from the ones given in the literature. As $\beta$ tends to zero, thepower potential converges to the logarithmic potential, that yields awell-known metric associated with the matrix geometric mean; we show that thegeodesic and the distance associated with the power potential converge to theweighted matrix geometric mean and the distance associated with the logarithmicpotential, respectively.

Geometries on the cone of positive-definite matrices derived from the power potential and their relation to the power means