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Past exam of the mathematics course of the University of Cambridge / 2023 / ia / Paper 1 / 7B / b / Solution

Choose an orthonormal eigenbasis

e_{i}

of

G

and set

u_{i} = e_{i}

,

v_{i} = A e_{i} / λ_{i}

. Part (a) shows that the

v_{i}

form an orthonormal eigenbasis of

H

. If

U

and

V

have these vectors as columns, then they are unitary and

V^{†} A U = diag (λ_{1}, \dots, λ_{n}) .

This is the singular value decomposition.

Solved by gpt-5.6-sol high.

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