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, $$ \begin{pmatrix} 1 & 0 \\ 0 & 0 \\ 0 & 0 \end{pmatrix},\quad \begin{pmatrix} 0 & 1 \\ 0 & 0 \\ 0 & 0 \end{pmatrix},\quad \begin{pmatrix} 0 & 0 \\ 1 & 0 \\ 0 & 0 \end{pmatrix},\quad \begin{pmatrix} 0 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix},\quad \begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 1 & 0 \end{pmatrix},\quad \begin{pmatrix} 0 & 0 \\ 0 & 0 \\ 0 & 1 \end{pmatrix} $$ 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. $AU_{ij}$ 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:
  1. $U_{ij}U_{k\ell}=\delta_{jk}U_{i\ell}$ , and
  2. $U_{11}+\cdots+U_{nn}=I_n$ ,
  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 $i\ne j$ .
• 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 $\lbrace U_{ij}\mid 1\le i,j\le 2\rbrace$ is the set of $2\times 2$ matrix units over $\mathbb{R}$ , then for any invertible matrix $T$ , $\lbrace TU_{ij}T^{-1}\mid 1\le i,j\le 2\rbrace$ 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 $R$ .

Bibliography

1

T. Y. Lam, Lectures on Modules and Rings, Springer, New York, 1998.