Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 225 / 3 / a / Solution

For positive trace-class operators, the square-root distance between covariance operators isIf $C_i=L_i{L}_i^{\ast }$ are Hilbert-Schmidt factorizations, the Procrustes distance between covariance operators iswhere $\mathcal{O}(H)$ is the group of orthogonal operators on the real separable Hilbert space $H$.