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 <math>E_{ij}</math>. For example, the 3 by 3 matrix unit with i = 1 and j = 2 is
<math display=block>E_{12} = \begin{bmatrix}0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}</math>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.
: <math>
E_{23}A = \left[ \begin{matrix} 0 & 0& 0 \\ a_{31} & a_{32} & a_{33} \\ 0 & 0 & 0 \end{matrix}\right].
</math>
: <math>
AE_{23} = \left[ \begin{matrix} 0 & 0 & a_{12} \\ 0 & 0 & a_{22} \\ 0 & 0 & a_{32} \end{matrix}\right].
</math>