elementary matrix
Elementary Operations on Matrices
Let $\mathbb{M}$ be the set of all $m\times n$ matrices (over some commutative ring $R$). An operation^{} on $\mathbb{M}$ is called an elementary row operation if it takes a matrix $M\in \mathbb{M}$, and does one of the following:

1.
interchanges of two rows of $M$,

2.
multiply a row of $M$ by a nonzero element of $R$,

3.
add a (constant) multiple of a row of $M$ to another row of $M$.
An elementary column operation is defined similarly. An operation on $\mathbb{M}$ is an elementary operation if it is either an elementary row operation or elementary column operation.
For example, if $M=\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \\ \hfill e\hfill & \hfill f\hfill \end{array}\right)$, then the following operations correspond respectively to the three types of elementary row operations described above

1.
$\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill e\hfill & \hfill f\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)$ is obtained by interchanging rows 2 and 3 of $M$,

2.
$\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill rc\hfill & \hfill rd\hfill \\ \hfill e\hfill & \hfill f\hfill \end{array}\right)$ is obtained by multiplying $r\ne 0$ to the second row of $M$,

3.
$\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \\ \hfill sa+e\hfill & \hfill sb+f\hfill \end{array}\right)$ is obtained by adding to row 1 multiplied by $s$ to row 3 of $M$.
Some immediate observation: elementary operations of type 1 and 3 are always invertible^{}. The inverse^{} of type 1 elementary operation is itself, as interchanging of rows twice gets you back the original matrix. The inverse of type 3 elementary operation is to add the negative of the multiple of the first row to the second row, thus returning the second row back to its original form. Type 2 is invertible provided that the multiplier has an inverse.
Some notation: for each type $k$ (where $k=1,2,3$) of elementary operations, let ${E}_{c}^{k}(A)$ be the set of all matrices obtained from $A$ via an elementary column operation of type $k$, and ${E}_{r}^{k}(A)$ the set of all matrices obtained from $A$ via an elementary row operation of type $k$.
Elementary Matrices
Now, assume $R$ has $1$. An $n\times n$ elementary matrix is a (square) matrix obtained from the identity matrix^{} ${I}_{n}$ by performing an elementary operation. As a result, we have three types of elementary matrices, each corresponding to a type of elementary operations:

1.
transposition matrix ${T}_{ij}$: an matrix obtained from ${I}_{n}$ with rows $i$ and $j$ switched,

2.
basic diagonal matrix ${D}_{i}(r)$: a diagonal matrix^{} whose entries are $1$ except in cell $(i,i)$, whose entry is a nonzero element $r$ of $R$

3.
row replacement matrix ${E}_{ij}(s)$: ${I}_{n}+s{U}_{ij}$, where $s\in R$ and ${U}_{ij}$ is a matrix unit with $i\ne j$.
For example, among the $3\times 3$ matrices, we have
$${T}_{12}=\left(\begin{array}{ccc}\hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right),{D}_{3}(r)=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill r\hfill \end{array}\right),\text{and}\mathit{\hspace{1em}}{E}_{32}(s)=\left(\begin{array}{ccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill s\hfill & \hfill 1\hfill \end{array}\right)$$ 
For each positive integer $n$, let ${\mathbb{E}}^{k}(n)$ be the collection^{} of all $n\times n$ elementary matrices of type $k$, where $k=1,2,3$.
Below are some basic properties of elementary matrices:

•
${T}_{ij}={T}_{ji}$, and ${T}_{ij}^{2}={I}_{n}$.

•
${D}_{i}(r){D}_{i}({r}^{1})={I}_{n}$, provided that ${r}^{1}$ exists.

•
${E}_{ij}(s){E}_{ij}(s)={I}_{n}$.

•
$det({T}_{ij})=1$, $det({D}_{i}(r))=r$, and $det({E}_{ij}(s))=1$.

•
If $A$ is an $m\times n$ matrix, then
$${E}_{c}^{k}(A)=\{AE\mid E\in {\mathbb{E}}^{k}(n)\}\mathit{\hspace{1em}\hspace{1em}}\text{and}\mathit{\hspace{1em}\hspace{1em}}{E}_{r}^{k}(A)=\{EA\mid E\in {\mathbb{E}}^{k}(m)\}.$$ 
•
Every nonsingular matrix can be written as a product^{} of elementary matrices. This is the same as saying: given a nonsingular matrix, one can perform a finite number of elementary row (column) operations on it to obtain the identity matrix.
Remarks.

•
One can also define elementary matrix operations on matrices over general rings. However, care must be taken to consider left scalar multiplications and right scalar multiplications as separate operations.

•
The discussion above pertains to elementary linear algebra. In algebraic Ktheory, an elementary matrix is defined only as a row replacement matrix (type 3) above.
Title  elementary matrix 
Canonical name  ElementaryMatrix 
Date of creation  20130322 18:30:38 
Last modified on  20130322 18:30:38 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  13 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 1501 
Related topic  MatrixUnit 
Related topic  GaussianElimination 
Defines  elementary operation 
Defines  elementary column operation 
Defines  elementary row operation 
Defines  basic diagonal matrix 
Defines  transposition matrix 
Defines  row replacement matrix 