22. Elementary matrices, 1

0.00 We introduce elementary matrices which have the following property: multiplying $A$ on the left by an elementary matrix has the effect of performing a row operation to $A$ .

Elementary matrices of type I

Definition:

0.25 Let $i, j$ be integers between $1$ and $n$ inclusive, and assume $i\neq j$ . Let $\lambda\in\mathbf{R}$ . Then $E_{ij}(\lambda)$ is the matrix which has:

1s on the diagonal,
0s everywhere else except in the $i j$ th position, where there is a $\lambda$ .

Such a matrix is an elementary matrix of type I.

1.50 For example, if $n=3$ you get things like: $E_{12}(4)=\begin{pmatrix}1&4&0\\ 0&1&0\\ 0&0&1\end{pmatrix},\qquad E_{32}(-5)=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&-5&1\end{pmatrix}.$ 2.48 If $n=4$ , you get things like $E_{23}(7)=\begin{pmatrix}1&0&0&0\\ 0&1&7&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}.$

Lemma:

4.05 Let $A$ and $E_{ij}(\lambda)$ be $n$ -by- $n$ matrices. Then $E_{ij}(\lambda)A$ is the matrix obtained from $A$ by doing $R_{i}\mapsto R_{i}+\lambda R_{j}$ .

5.16 Let's consider the case $i<j$ (the other case is similar so we omit it). Consider the product $\begin{pmatrix}1&&\mbox{col }i&&\mbox{col }j&&\\ &\ddots&\downarrow&&\downarrow&&\\ \mbox{row }i&\rightarrow&1&&\lambda&&\\ &&&\ddots&&&\\ &&&&1&&\\ &&&&&\ddots&\\ &&&&&&1\end{pmatrix}\begin{pmatrix}A_{11}&\cdots&\cdots&&\cdots&\cdots&A_{1n}% \\ \vdots&&&&&&\vdots\\ A_{i1}&&&&&&A_{in}\\ \vdots&&&&&&\vdots\\ A_{j1}&&&&&&A_{jn}\\ \vdots&&&&&&\vdots\\ A_{n1}&\cdots&\cdots&&\cdots&\cdots&A_{nn}\end{pmatrix}$ 7.05 The only difference the $\lambda$ makes is when we multiply the $i$ th row into a column of $A$ (say the $k$ th column). Instead of just picking up $1\times A_{ik}$ , we get $1\times A_{ik}+\lambda\times A_{jk}$ .

8.56 In other words, the result $E_{ij}(\lambda)A$ is obtained from $A$ by adding $\lambda$ times row $j$ to row $i$ .

Elementary matrices of type II

Definition:

9.24 Let $i$ be an integer between $1$ and $n$ inclusive and let $\lambda\neq 0$ be a real number. Define $E_{i}(\lambda)$ to be the matrix which has 1s on the diagonal everywhere except in the $i i$ th position, where it has a $\lambda$ . Such a matrix is an elementary matrix of type II.

10.19 If $n=3$ , $E_{2}(\lambda)=\begin{pmatrix}1&0&0\\ 0&\lambda&0\\ 0&0&1\end{pmatrix},\qquad E_{3}(-1)=\begin{pmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{pmatrix}.$

Lemma:

11.00 $E_{i}(\lambda)A$ is obtained from $A$ by performing the row operation $R_{i}\mapsto\lambda R_{i}$ .

In the product, $\begin{pmatrix}1&0&\cdots&&0\\ 0&\ddots&&&\\ \vdots&&\lambda&&\vdots\\ &&&\ddots&0\\ 0&&\cdots&0&1\end{pmatrix}\begin{pmatrix}A_{11}&&\cdots&&A_{1n}\\ \vdots&&&&\vdots\\ A_{i1}&&\cdots&&A_{in}\\ \vdots&&&&\vdots\\ A_{n1}&&\cdots&&A_{nn}\end{pmatrix}$ most of the entries are multiplied by $1$ , except those in the $i$ th row of $A$ , which are rescaled by $\lambda$ .