 Home

Codex Wiki

List of universities

University of Cambridge

Course of the University of Cambridge

Mathematics course of the University of Cambridge

Past exam of the mathematics course of the University of Cambridge

2022

Paper 225

For an orthogonal $R$, unitary invariance of the <Hilbert-Schmidt norm> gives
$$
\lVert L_1-L_2R\rVert_{\rm HS}^2
=\lVert L_1\rVert_{\rm HS}^2+\lVert L_2\rVert_{\rm HS}^2
-2\operatorname{Re}\operatorname{tr}(R^*L_2^*L_1).
$$
The product $L_2^*L_1$ is trace-class. Its <polar decomposition of a bounded operator> and <trace duality> give
$$
\sup_{R\in\mathcal O(H)}
\operatorname{Re}\operatorname{tr}(R^*L_2^*L_1)
=\lVert L_2^*L_1\rVert_1
=\sum_{k=1}^{\infty}\sigma_k,
$$
where $(\sigma_k)$ are its <singular values>. Taking the infimum proves
$$
d_P(C_1,C_2)^2
=\lVert L_1\rVert_{\rm HS}^2+\lVert L_2\rVert_{\rm HS}^2
-2\sum_{k=1}^{\infty}\sigma_k.
$$

Solution

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

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

 Ancestors (11)

 Discussion (0)

 Articles by others on the same topic (0)

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

 Ancestors (11)

 Discussion (0)  Subscribe (1)

 Articles by others on the same topic (0)

 Discussion (0)