Matrix Unit

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 ''i''th row and ''j''th column is denoted as $E_$. For example, the 3 by 3 matrix unit with ''i'' = 1 and ''j'' = 2 is
$E_ = \begin0 & 1 & 0 \\0 & 0 & 0 \\ 0 & 0 & 0 \end$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
$E_E_ = \delta_E_,$
where $\delta_$ 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''.

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_A = \left \begin 0 & 0& 0 \\ a_ & a_ & a_ \\ 0 & 0 & 0 \end\right

:
AE_ = \left \begin 0 & 0 & a_ \\ 0 & 0 & a_ \\ 0 & 0 & a_ \end\right

References

Sparse matrices
1 (number)

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