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According to this paper, every trace zero matrix over a field can be written in the form $AB - BA$. However, here's a basic counterexample:

Let $A = diag(a, -a)$ for some nonzero number a. Then $A = BC - CB$ if and only if a system of equations involving entries of $B,C,A$ is true. Then the entries of $A$ are:

$$\begin{align} a = b_{1,1}c_{1,1} + b_{1,2}c_{2,1} - b_{1,1}c_{1,1} - b_{2,1}c_{1,2} \\ 0 = b_{1,1}c_{1,2} + b_{1,2}c_{2,2} - b_{1,1}c_{2,1} - b_{2,1}c_{2,2} \\ -a = b_{2,1}c_{1,1} + b_{2,2}c_{2,1} - b_{1,2}c_{1,1} - b_{2,2}c_{1,2} \\ 0 = b_{2,1}c_{1,2} + b_{2,2}c_{2,2} - b_{1,2}c_{2,1} - b_{2,2}c_{2,2} \end{align} $$

For $a$ nonzero the first and the last equations can't both be true. So where did I mess up?

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  • $\begingroup$ qaphla, how did you do that so fast? $\endgroup$ – BananaCats Category Theory App Aug 12 '13 at 18:07
  • $\begingroup$ Lots of practice with LaTeX and I type about 85-90wpm. $\endgroup$ – qaphla Aug 12 '13 at 18:08
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    $\begingroup$ I believe you want to interchange -a and 0 in the last two equations. $\endgroup$ – user84413 Aug 12 '13 at 18:08
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    $\begingroup$ @user84413. And there it is! Thanks! $\endgroup$ – BananaCats Category Theory App Aug 12 '13 at 18:09
  • $\begingroup$ For reference: I think this open access 1957 paper of Albert and Muckenhoupt was the first published proof of this fact for a general field of any characteristic. $\endgroup$ – user85486 Aug 14 '13 at 2:11
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Consider $$\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\begin{bmatrix} 0 & 0 \\ a & 0 \end{bmatrix} -\begin{bmatrix} 0 & 0 \\ a & 0 \end{bmatrix}\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$$

What made you conclude that the first and last of those equations can't be true?

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  • $\begingroup$ I made a mistake, writing them down on paper. But qaphla corrected it. Two LHS's were swapped. $\endgroup$ – BananaCats Category Theory App Aug 12 '13 at 18:14
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$$B=\left(\begin{array}{rr}0&1\\0&0\end{array}\right).$$ $$C=a\left(\begin{array}{rr}0&0\\1&0\end{array}\right).$$ $$BC-CB=\left(\begin{array}{rr}a&0\\0&-a\end{array}\right).$$

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  • $\begingroup$ Why does your avatar say community wiki and have no rating? $\endgroup$ – BananaCats Category Theory App Aug 12 '13 at 18:12
  • $\begingroup$ meta.stackexchange.com/questions/11740/… $\endgroup$ – Jim Aug 12 '13 at 18:16
  • $\begingroup$ Citing Henning: "Credit waived". I use it occasionally. This time partly because I didn't address your question, i.e. I didn't help you spotting the mistake. $\endgroup$ – Jyrki Lahtonen Aug 12 '13 at 18:18

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