Let $X$ be $n\times n$ matrix. What are the necessary and sufficient conditions of the existence of representation: $X=AB-BA$ where $A,B$ are some $n\times n$ matrices. This seems difficult. How to start?
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Well $X$ has to have trace zero for starters. Hopefully others can chime in with more conditions. – Cameron L. Williams Jul 23 '15 at 03:44
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1I think I could not understand your question – Empty Jul 23 '15 at 03:44
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@CameronWilliams why? where is intuition – nerd Jul 23 '15 at 03:45
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1If $X = AB - BA$, then $$\operatorname{tr} X = \operatorname{tr}(AB - BA) = \operatorname{tr}(AB) - \operatorname{tr}(BA) = \operatorname{tr}(AB) - \operatorname{tr}(AB) = 0.$$ – Cameron L. Williams Jul 23 '15 at 03:47
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3See this answer by Bill:http://math.stackexchange.com/a/99356/274 – Mariano Suárez-Álvarez Jul 23 '15 at 03:59
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@MarianoSuárez-Alvarez thanks for the link, i deleted my answer – user148177 Jul 23 '15 at 04:08