If nilpotent matrix $A$ and $AB−BA$ commute,show that $AB$ is nilpotent.


Let $ A$ and $ B$ be $ n×n$ complex matrices.

If $ A$ is an 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.

Definite linear transformation $ T$ as $ T(B)=AB-BA$ .

If $ A$ is an 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.