Defining a particular equivalence relation on square matrices relating to Jordan Canonical Form

Question

Suppose that $K$ is an algebraically closed field and $M$ is the set of all $n \times n$ matrices ($n \geq 1$) over $K$. Consider the subset $J = \{m \in M \;| \; m \mbox{ is in Jordan Canonical Form}\}$. I am interested in coming up with an equivalence relation $\sim$ on $M$ with the property that $J$ contains precisely one element from each equivalence class of $\sim$. At first, I was considering defining this equivalence relation as $m_1 \sim m_2$ iff $m_1$ and $m_2$ are matrix representations of the same linear operator. However, I don't think that this works, since Jordan Canonical Forms are unique only up to permutation of the blocks.

Answer 1

Let $O$ be the zero matrix. Define $\sim$ on $M$ by this:

(a) Every $m\in J$, $m\not=O$ is equivalent to itself and no other element of $M$;

(b) $O$ is equivalent to itself, all the matrices of $M\setminus J$, and no others;

(c) Every $m\not\in J$ is equivalent to the zero matrix $O$ , and to all the other elements $n\not\in J$, and to no others.

It's not "interesting", of course.