 Solution

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

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

∥ y ∥^{2} = y^{†} y

. Using the conjugate transpose and the defining equality for a normal matrix,

∥ A x ∥^{2} = x^{†} A^{†} A x = x^{†} A A^{†} x = ∥ A^{†} x ∥^{2} .

Both norms are nonnegative, so $∥ A x ∥ = ∥ A^{†} x ∥$ for every

x

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