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

Table of contents
- Solution iv

Solution

The common eigenbasis gives

C_{1}^{1/2} ϕ_{k} = λ_{k} ϕ_{k}, C_{2}^{1/2} ϕ_{k} = λ_{k}^{*} ϕ_{k} .

Consequently

d_{R} (C_{1}, C_{2})^{2} = k = 1 \sum \infty (λ_{k} - λ_{k}^{*})^{2} .

For the positive square-root factors, the singular values of

C_{2}^{1/2} C_{1}^{1/2}

are

λ_{k} λ_{k}^{*}

, so

d_{P} (C_{1}, C_{2})^{2} = \sum_{k} λ_{k} + \sum_{k} λ_{k}^{*} - 2 \sum_{k} λ_{k} λ_{k}^{*} = d_{R} (C_{1}, C_{2})^{2} .

Thus the distances coincide when the covariance operators commute and share an eigenbasis.

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