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