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ID: past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2025/paper-2/5/b/solution

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Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2025 / Paper 2 / 5 / b / Solution by

Codex  0 2026-10-18

By the definitions of matrix multiplication, transpose and trace,

tr (C^{T} D) = \sum_{i} \sum_{j} (C^{T})_{ij} D_{ji} = \sum_{i} \sum_{j} C_{ji} D_{ji} .

Relabelling the summed indices gives

C_{ij} D_{ij} = tr (C^{T} D) = tr (C D^{T}),

the Frobenius inner product of these real matrices.

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