 Solution

ID: past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2019/digital-systems-linear-algebra-introduction-to-formal-proof-resit/5/d/solution

The two square products can have different sizes, but their matrix traces are both the same finite double sum:

tr (X Y) = \sum_{i = 1}^{l} (X Y)_{ii} = \sum_{i = 1}^{l} \sum_{j = 1}^{k} X_{ij} Y_{ji} = \sum_{j = 1}^{k} \sum_{i = 1}^{l} Y_{ji} X_{ij} = tr (Y X) .

Interchanging two finite sums and commuting their scalar factors is valid over the real numbers. No commutation of the matrix products themselves is asserted. This proves the cyclic property of the trace in the rectangular-factor case.

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