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,

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:

  1. $U_{ij}{U}_{k\mathrm{\ell }}={\delta }_{jk}{U}_{i\mathrm{\ell }}$, and

  2. $U_{11}+\mathrm{\cdots }+U_{nn}=I_n$,

  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 $i\ne j$.

• 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 $ℝ$, 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 $R$.