Matrix unit

Not to be confused with Unit matrix.

In linear algebra, a matrix unit is a matrix with only one nonzero entry with value 1. The matrix unit with a 1 in the ith row and jth column is denoted as $E_{ij}$. For example, the 3 by 3 matrix unit with i = 1 and j = 2 is $$E_{12}={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}}$$

Properties

The set of m by n matrix units is a basis of the space of m by n matrices.

The product of two matrix units of the same square shape $n\times n$ satisfies the relation $$E_{ij}E_{kl}=\delta _{jk}E_{il},$$ where $\delta _{jk}$ is the Kronecker delta.

The group of scalar n-by-n matrices over a ring R is the centralizer of the subset of n-by-n matrix units in the set of n-by-n matrices over R.