# 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 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.

• Compare also with this question. Commented Apr 17, 2019 at 19:27
• This is actually the main theorem of an article by Kaplansky. The statement is even generalized there. Commented Apr 18, 2019 at 21:34
• @Helmut: The link is dead. What article of Kaplansky's is it? Maybe "Jacobson's Lemma Revisited" (J. Algebra 62, 473-476 (1980)), cf. zbmath.org/?q=an%3A0425.16020 ? Commented Apr 5, 2021 at 3:36
• @TorstenSchoeneberg Yes, that is the article. By the way, the link is active again:-) Commented Apr 10, 2021 at 13:59