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 $r\eq 0$ 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 $i\eq j$ .

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.