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.^[1][2] The matrix unit with a 1 in the ith row and jth column is denoted as ${\displaystyle E_{ij}}$. For example, the 3 by 3 matrix unit with i = 1 and j = 2 is $${\displaystyle E_{12}={\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}}}$$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.^[2]

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.^[2]

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.^[2]

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:^[3]

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

${\displaystyle AE_{23}=\left[{\begin{matrix}0&0&a_{12}\\0&0&a_{22}\\0&0&a_{32}\end{matrix}}\right].}$

References

1. Artin, Michael. Algebra. Prentice Hall. p. 9.
2. Lam, Tsit-Yuen (1999). "Chapter 17: Matrix Rings". Lectures on Modules and Rings. Graduate Texts in Mathematics. Vol. 189. Springer Science+Business Media. pp. 461–479.
3. Marcel Blattner (2009). "B-Rank: A top N Recommendation Algorithm". arXiv:0908.2741 [physics.data-an].