The Fréchet distance between multivariate normal distributions

The Fréchet distance between two multivariate normal distributions having means [mu]X, [mu]Y and covariance matrices [Sigma]X, [Sigma]Y is shown to be given by d2 = [mu]X - [mu]Y2 + tr([Sigma]X + [Sigma]Y - 2([Sigma]X[Sigma]Y)1/2). The quantity d0 given by d02 = tr([Sigma]X + [Sigma]Y - 2([Sigma]X[Sigma]Y)1/2) is a natural metric on the space of real covariance matrices of given order.