Matrix unit

The set of m by n matrix units is a basis of the space of m by n matrices.[2]

The product of two matrix units of the same square shape ${\displaystyle n\times n}$ satisfies the relation

$${\displaystyle E_{ij}E_{kl}=\delta _{jk}E_{il},}$$

where ${\displaystyle \delta _{jk}}$ is the Kronecker delta.[2]

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.[2]

The matrix norm (induced by the same two vector norms) of a matrix unit is equal to 1.

When multiplied by another matrix, it isolates a specific row or column in arbitrary position. For example, for any 3-by-3 matrix A:[3]

${\displaystyle E_{23}A=\left[{\begin{matrix}0&0&0\\a_{31}&a_{32}&a_{33}\\0&0&0\end{matrix}}\right].}$

${\displaystyle AE_{23}=\left[{\begin{matrix}0&0&a_{12}\\0&0&a_{22}\\0&0&a_{32}\end{matrix}}\right].}$