 Solution

ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2014/12/iii/solution

tr (A B) = tr (B A)

, which is C1. The linearity of trace and matrix multiplication gives C2 and C3. Finally symmetry of

A

gives

⟨ A, A ⟩ = tr (A^{2}) = \sum_{i, j} A_{ij} A_{ji} = \sum_{i, j} A_{ij}^{2} \geq 0,

with equality exactly when every entry is zero. Thus C4 also holds. This is the Frobenius inner product restricted to the real vector space of symmetric matrices.

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