Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 225 / 3 / iv / Solution

The common eigenbasis givesConsequentlyFor the positive square-root factors, the singular values of $C_2^{1/2}{C}_1^{1/2}$ are $\sqrt{{\lambda }_k{\lambda }_k^{\ast }}$, soThus the distances coincide when the covariance operators commute and share an eigenbasis.