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 non-zero 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 $M=\begin{pmatrix}a&b\\ c&d\\ e&f\end{pmatrix}$ , then the following operations correspond respectively to the three types of elementary row operations described above

1.

$\begin{pmatrix}a&b\\ e&f\\ c&d\end{pmatrix}$ is obtained by interchanging rows 2 and 3 of $M$ ,
2.

$\begin{pmatrix}a&b\\ rc&rd\\ e&f\end{pmatrix}$ is obtained by multiplying $r\neq 0$ to the second row of $M$ ,
3.

$\begin{pmatrix}a&b\\ c&d\\ sa+e&sb+f\end{pmatrix}$ 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 non-zero element $r$ of $R$
3.

row replacement matrix $E_{ij}(s)$ : $I_{n}+sU_{ij}$ , where $s\in R$ and $U_{ij}$ is a matrix unit with $i\neq j$ .

For example, among the $3\times 3$ matrices, we have

$T_{12}=\begin{pmatrix}0&1&0\\ 1&0&0\\ 0&0&1\end{pmatrix},\quad D_{3}(r)=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&r\end{pmatrix},\quad\mbox{and}\quad E_{32}(s)=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&s&1\end{pmatrix}$

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

$E_{c}^{k}(A)=\{AE\mid E\in\mathbb{E}^{k}(n)\}\qquad\mbox{and}\qquad E_{r}^{k}(% A)=\{EA\mid E\in\mathbb{E}^{k}(m)\}.$

•

Every non-singular matrix can be written as a product of elementary matrices. This is the same as saying: given a non-singular 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 K-theory, an elementary matrix is defined only as a row replacement matrix (type 3) above.

Title	elementary matrix
Canonical name	ElementaryMatrix
Date of creation	2013-03-22 18:30:38
Last modified on	2013-03-22 18:30:38
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	13
Author	CWoo (3771)
Entry type	Definition
Classification	msc 15-01
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