Correct way to denote set of matrices with one $1$ and one $-1$ on the diagonal, zeros elsewhere.

Question

I am trying to describe the set of roots of a particular Lie algebra representation, and I was looking for suggestions to "clean up" my notation a bit. The roots are $n\times n$ matrices having a $1$ and a $-1$ in the diagonal, and right now I've got something like this:

$$\mathcal{R}=\{(a_{ii})-(a_{jj}):(a_{kk})= \begin{pmatrix}0\\&\ddots\\&&1\\&&&\ddots\\&&&&0\end{pmatrix}\text{, having }1\text{ in the }kk\text{ position.}\}$$

Is there a better way to notate this?

Answer 1

A fairly common notation is to use matrix units: that is, $E_{kj}$ is the matrix with a $1$ in the $k,j$ entry, and zeroes elsewhere. Then you would have $$ \mathcal R=\{E_{kk}-E_{jj}:\ k\ne j\}. $$