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

Choose a unit vector $e$ and distinct positive numbers $a,b$, and setThese are distinct rank-one covariance operators. Their positive square roots are $\sqrt{a}(e\otimes e)$ and $\sqrt{b}(e\otimes e)$, soUsing these square roots as the factors in the Procrustes definition, the identity alignment attains the same value. The singular-value formula gives the matching lower bound,Hence $d_P=d_R$ although $C_1\ne C_2$.