Is characterisation of degree 2 nilpotent matrices (i.e. $M^2=0$) known?

Question

$M$ is $n\times n$ real (or complex) matrix. Also $M$ is nilpotent of degree 2, i.e. $M^2=0.$

Question. How does $M$ look like?

I just calculated that $2\times 2$ matrix must have following form $$\begin{bmatrix} gh & \pm g^2 \\ \mp h^2 & -gh \end{bmatrix}.$$

I wanted to compute conditions on $3\times3,4\times 4$ and look for some pattern, but I thought that such problem should have been done long time ago. So

Or just reference request. Are there any sources that deals with this problem?

If one wants to know the origin, then this problem is related to this my unsolved problem.

PS. I added (homological-algebra) tag, cause of condition $M^2=0.$

Answer 1

Note that for any matrix $M$ satisfying $M^2 = 0$ and any invertible $S$, the matrix $N = SMS^{-1}$ satisfies $N^2 = 0$. Thus, it is common when answering a question like this to solve it "up to similarity". That is, provide a single element from each conjugacy class.

If we suppose that $M$ is in Jordan canonical form: every Jordan block is a block associated with zero, and the maximum size is $2$. That is, there is one conjugacy class for each distinct partition of the form $n = 2+2+\dots+2+1+1+\dots+1$.

If we instead suppose $M$ is in Weyr canonical form, then $M$ has the form $$ M = \pmatrix{ 0_{k \times k}&0&I_{k \times k}\\ 0&0_{(n-2k) \times (n - 2k)}&0\\ 0&0&0_{k\times k}} $$ With $k = 0,1,2,\dots,\lfloor n/2\rfloor$.

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Similarly: for any polynomial $p(x)$, it is easy to find the $M$ satisfying $p(M) = 0$ up to similarity. One useful reference is chapter 3 of Horn and Johnson's Matrix Analysis.