matrix unit
A matrix unit is a matrix (over some ring with $1$) whose entries are all $0$ except in one cell, where it is $1$.
For example, among the $3\times 2$ matrices,
$$\left(\begin{array}{cc}\hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),\left(\begin{array}{cc}\hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill \end{array}\right),\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill \end{array}\right),\left(\begin{array}{cc}\hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right)$$ 
are the matrix units.
Let $A$ and $B$ be $m\times n$ and $p\times q$ matrices over $R$, and ${U}_{ij}$ an $n\times p$ matrix unit (over $R$). Then

1.
$A{U}_{ij}$ is the $m\times p$ matrix whose $j$th column is the $i$th column of $A$, and $0$ everywhere else, and

2.
${U}_{ij}B$ is the $n\times q$ matrix whose $i$th row is the $j$th row of $B$ and $0$ everywhere else.
Remarks. Let $M={M}_{m\times n}(R)$ be the set of all $m$ by $n$ matrices with entries in a ring $R$ (with $1$). Denote ${U}_{ij}$ the matrix unit in $M$ whose cell $(i,j)$ is $1$.

•
$M$ is a (left or right) $R$module generated by the $m\times n$ matrix units.

•
When $m=n$, $M$ has the structure of an algebra over $R$. The matrix units have the following properties:

(a)
${U}_{ij}{U}_{k\mathrm{\ell}}={\delta}_{jk}{U}_{i\mathrm{\ell}}$, and

(b)
${U}_{11}+\mathrm{\cdots}+{U}_{nn}={I}_{n}$,
where ${\delta}_{ij}$ is the Kronecker delta^{} and ${I}_{n}$ is the identity matrix^{}. Note that the ${U}_{ii}$ form a complete set of pairwise orthogonal idempotents, meaning ${U}_{ii}{U}_{ii}={U}_{ii}$ and ${U}_{ii}{U}_{jj}=0$ if $i\ne j$.

(a)

•
In general, in a matrix ring $S$ (consisting of, say, all $n\times n$ matrices), any set of $n$ matrices satisfying the two properties above is called a full set of matrix units of $S$.

•
For example, if $\{{U}_{ij}\mid 1\le i,j\le 2\}$ is the set of $2\times 2$ matrix units over $\mathbb{R}$, then for any invertible matrix $T$, $\{T{U}_{ij}{T}^{1}\mid 1\le i,j\le 2\}$ is a full set of matrix units.

•
If we embed $R$ as a subring of ${M}_{n}(R)$, then $R$ is the centralizer^{} of the matrix units of ${M}_{n}(R)$, meaning that the only elements in ${M}_{n}(R)$ that commute with the matrix units are the elements in $R$.
References
 1 T. Y. Lam, Lectures on Modules and Rings, Springer, New York, 1998.
Title  matrix unit 

Canonical name  MatrixUnit 
Date of creation  20130322 18:30:35 
Last modified on  20130322 18:30:35 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  8 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 15A30 
Classification  msc 16S50 
Related topic  ElementaryMatrix 
Defines  full set of matrix units 