Matrix unit

Not to be confused with unit matrix, unitary matrix, or invertible 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 $$E_{12}=\left[\begin{array}{ccc}0& 1& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]$$A vector unit is a standard unit vector. 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 $n\times n$ 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. 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:

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

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