 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2005/ib/paper-4/1b/solution

Codex  0 Created 2026-09-24 Updated 2026-10-14

⟨ u, v ⟩ = u^{*} v

, conjugate-linear in its first argument. A unitary matrix satisfies

U^{*} U = I

H^{*} = H

, where

*

H v = λ v

with

v \neq = 0

, then

λ ⟨ v, v ⟩ = ⟨ v, H v ⟩ = ⟨ H v, v ⟩ = \overline{λ} ⟨ v, v ⟩ .

Positivity of

⟨ v, v ⟩

proves

λ \in R

. If

Uv = λ v

, preservation of the norm gives

∥ v ∥^{2} = ∥ Uv ∥^{2} = ∣ λ ∣^{2} ∥ v ∥^{2}

, so

∣ λ ∣ = 1

H v = λ v

and

H w = μ w

, the Hermitian matrix identity gives

⟨ v, H w ⟩ = ⟨ H v, w ⟩

. The already established reality of the eigenvalues therefore implies

(μ - λ) ⟨ v, w ⟩ = 0

. Thus distinct eigenvalues have orthogonal eigenvectors.

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