Matrix unit

A single-entry matrix generalizes the matrix unit for matrices with only one nonzero entry of any value, not necessarily of value 1.

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 ${\displaystyle n\times n}$ satisfies the relation $${\displaystyle E_{ij}{E}_{kl}={\delta }_{jk}{E}_{il},}$$ where ${\displaystyle {\delta }_{jk}}$ 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.

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:

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

${\displaystyle A{E}_{23}=\left[\begin{array}{ccc}0& 0& a_{12}\\ 0& 0& a_{22}\\ 0& 0& a_{32}\end{array}\right].}$