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 i j {\displaystyle E_{ij}} . For example, the 3 by 3 matrix unit with i = 1 and j = 2 is
E 12 = [ 0 1 0 0 0 0 0 0 0 ] {\displaystyle E_{12}={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}}} 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 × n {\displaystyle n\times n} satisfies the relation
E i j E k l = δ j k E i l , {\displaystyle E_{ij}E_{kl}=\delta _{jk}E_{il},}
where δ j k {\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:
E 23 A = [ 0 0 0 a 31 a 32 a 33 0 0 0 ] . {\displaystyle E_{23}A=\left[{\begin{matrix}0&0&0\\a_{31}&a_{32}&a_{33}\\0&0&0\end{matrix}}\right].}
A E 23 = [ 0 0 a 12 0 0 a 22 0 0 a 32 ] . {\displaystyle AE_{23}=\left[{\begin{matrix}0&0&a_{12}\\0&0&a_{22}\\0&0&a_{32}\end{matrix}}\right].}
References
