 Solution

ID: past-exam-of-the-mathematics-course-of-the-year-2-of-the-natural-sciences-course-of-the-university-of-cambridge/2024/paper-1/5/a/solution

Let

A v = λ v

with

v \neq = 0

v

gives

v^{†} A v = λ ∥ v ∥^{2}

\overline{v^{†} A v} = v^{†} A^{†} v = - v^{†} A v .

Consequently

\overline{λ} ∥ v ∥^{2} = - λ ∥ v ∥^{2}

∥ v ∥^{2} > 0

, the eigenvalue satisfies

\overline{λ} = - λ

, or

λ = i θ with θ \in R

. Zero is included as a purely imaginary eigenvalue.

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