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Trace of a product

MediumLinear Algebra~3m

For conformable matrices AA and BB, is tr(AB)=tr(BA)\mathrm{tr}(AB) = \mathrm{tr}(BA) always true (even though AB≠BAAB \ne BA in general)?

Show hints (2)+
  1. Write the trace as a double sum.
  2. It is the cyclic property of trace.

Answer

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Yes, always — both equal ∑i,jAijBji\sum_{i,j} A_{ij}B_{ji}

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Asked at: Jane Street, Two Sigma

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