Let $A$ and $B$ be $n×n$ complex matrices.
If $A$ is a nilpotent matrix, and $A$ commute with $AB−BA$, show that $AB$ is nilpotent.
Equivalently, the question can be expressed as following description.
Let $A$ and $B$ be $n×n$ complex matrices.
Define the linear transformation $T$ as $T(B)=AB-BA$.
If $A$ is a nilpotent matrix, and $T^2(B)=0$ , show that $AB$ is nilpotent.
I've known that $AB-BA$ is nilpotent.
Furtherly, if $A^m=0$ , by considering $T^n(B)=\sum_{i=0}^n(-1)^iA^{n-i}BA^i$ , I found that $A^kBA^l=0$ when $k+l\geqslant m$.
But I don't know how to continue, thanks for any help.
