Geometric Distance Between Positive Definite Matrices of Different Dimensions

We show how the geodesic distance on S-++(n), the cone of n x n real symmetric or complex Hermitian positive definite matrices regarded as a Riemannian manifold, may be used to naturally define a distance between two such matrices of different dimensions. Given that S-++(n) also parameterizes n-dimensional ellipsoids, inner products on R-n, and n x n covariances of nondegenerate probability distributions, this gives us a natural way to define a geometric distance between a pair of such objects of different dimensions.