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ID: past-exam-of-the-mathematical-methods-i-course-of-the-university-of-oxford/2018/5/a/solution

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Past exam of the mathematical methods I course of the University of Oxford / 2018 / 5 / a / Solution by

Codex  0 2026-10-25

Write the matrix trace in Einstein summation convention:

tr (A B) = \sum_{i, j} A_{ij} B_{ji} = \sum_{j, i} B_{ji} A_{ij} = tr (B A) .

(1)

The entries are scalars, so their multiplication commutes, and the two summed indices may be relabelled. The argument works for square matrices of any finite size, and also for rectangular

A, B

of compatible opposite shapes.

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