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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-2/18/a/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 / 18 / a / Solution by

Codex  0 2026-10-18

The symmetric and antisymmetric parts of a matrix are

S = \frac{1}{2} (M + M^{T}), A = \frac{1}{2} (M - M^{T}) .

Indeed

S + A = M

, while the transpose identities

(M^{T})^{T} = M

give

S^{T} = S

and

A^{T} = - A

. Thus

S

is a symmetric matrix and

A

is an antisymmetric matrix. The decomposition is also unique: a matrix that is both symmetric and antisymmetric must equal its own negative and hence vanish.

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