 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ia/paper-1/8a/a/iv/solution

Codex  0 Created 2026-09-24 Updated 2026-10-06

A^{†} x_{λ} = \overline{λ} x_{λ}

. Hence

μ x_{λ}^{†} x_{μ} = x_{λ}^{†} A x_{μ} = (A^{†} x_{λ})^{†} x_{μ} = λ x_{λ}^{†} x_{μ} .

μ \neq = λ

, $x_{λ}^{†} x_{μ} = 0$ . Thus eigenvectors in distinct eigenspaces of a normal matrix are orthogonal in the complex inner product.

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