Let $A$ and $B$ be $k\times k$ matrices such that $AB^k - B^kA = B$. I have to prove that $B$ is nilpotent. It is easy to see that $\text{trace}(B) = 0$. I do not have any idea how to proceed further.
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1Well you could try to prove that $\text{trace}(B^2)=0$ as well, etc – ancient mathematician Sep 10 '20 at 09:33
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1https://math.stackexchange.com/questions/159167/traces-of-all-positive-powers-of-a-matrix-are-zero-implies-it-is-nilpotent?noredirect=1&lq=1 – Sumanta Sep 10 '20 at 09:38
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Using properties of the trace and the characterization of $B$, you can show that: $$trace(B^n)=0 \quad \forall n>0$$ which implies that $B$ is nilpotent as shown here: Traces of all positive powers of a matrix are zero implies it is nilpotent
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