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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-1/6/a/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 / 6 / a / Solution by

Codex  0 2026-10-18

For every vector

v

, the Hermitian inner product gives

∥ C v ∥^{2} = v^{†} C^{†} C v = 0.

The norm vanishes only on the zero vector, so

C v = 0

for every

v

and

C = 0

. In contrast,

N = (0010) \neq = 0, N^{2} = 0

is a nonzero nilpotent matrix. Multiplying a matrix by its conjugate transpose detects column norms; multiplying it by itself has no corresponding positivity property.

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