Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a set of matrices that satisfy all of the following constraints?

1) $A^2=0, B^2=0, C^2=0...$ where $A,B,C,D..$ are different matrices.

2) All of them commute.

Edit: 3) $AB \neq 0, AC \neq 0, BC \neq 0, ...$.

share|cite|improve this question
What about all entries 0 except an $a$ in the top right corner, $a$ ranging over the elements of your field. – Jason Polak Oct 28 '12 at 0:26
edited. Forgot to mention 3). – TTTY Oct 28 '12 at 0:47
My answer to this earlier question of yours works here too. – Henning Makholm Oct 28 '12 at 1:05

Let $$E=\left(\begin{array}{cc} 0 & 1\\ 0 & 0 \end{array}\right), \ \ \ \ I=\left(\begin{array}{cc} 1 & 0\\ 0 & 1 \end{array}\right).$$

Then set

$$ A = E\otimes I\otimes I\otimes I,\\ B = I\otimes E\otimes I\otimes I,\\ C = I\otimes I\otimes E\otimes I,\\ D = I\otimes I\otimes I\otimes E. $$

Where $\otimes$ is the Kronecker product. These were constructed following the procedure outline by @HenningMakholm here:

More generally if only you have an $A$ and $B$ with $A^2=B^2$, $AB=BA=0$, you can combine $n$ copies of this structure using Kronecker products. That automatically makes matrices from different pairs commute, and ensures that cross-level products are nonzero as long as the generators are.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.