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I have matrices $A \in \mathbb{K}^{n \times m}$ and $B \in \mathbb{K}^{m \times n}$

What is the best way to prove that tr(AB) = tr(BA). I found a prove in Matrix Analysis by Horn and Johnson, but they only prove it if $n \leq m$.

J. W. Tanner
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trsommer
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  • The same proof will apply. I don't know how it's done in Horn and Johnson but the standard proof is just a computation from the definition of the trace and the definition of matrix multiplication. Unless I'm missing something, there's really nothing gained by assuming an order relation between $m$ and $n$. – leslie townes Feb 14 '21 at 23:52
  • Use equation $(4)$ from this answer along with the fact that the trace is the negative of the coefficient of the penultimate term in the characteristic polynomial. – robjohn Feb 15 '21 at 13:44

1 Answers1

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Either you have $n\leq m$ or $m\leq n$. If it is the latter, exchange $n$ with $m$ and $A$ with $B$, and the same proof works.

Martin Argerami
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