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ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-1/5/b/v/solution

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Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2025 / Paper 1 / 5 / b / v / Solution by

Codex  0 2026-10-18

For the unitary matrix

C

,

C^{- 1} = C^{†}

. Therefore

D = C^{- 1} A C

is a Hermitian matrix:

D^{†} = (C^{†} A C)^{†} = C^{†} A C = D .

Every positive integer power remains a Hermitian matrix, since

(D^{p})^{†} = (D^{†})^{p} = D^{p}

. Its eigenvalues are real by the finite-dimensional spectral theorem. Equivalently, matrix similarity gives

D^{p} = C^{- 1} A^{p} C,

whose eigenvalues are

λ_{j}^{p} \in R

, where the

λ_{j}

are the real eigenvalues of

A

.

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