---- > [!definition] Definition. ([[elementary operation]]) > > The following three operations on a [[matrix]] $P$ with entries in [[ring]] $R$ are called **elementary operations**. Each belongs to the [[general linear group]] over $R$. > > 1. Switch two rows (or two columns) of $P$; > 2. Add to one row (resp. column) a multiple of another row (resp. column); > 3. Multiply all entries in one row (or column) of $P$ by a [[unit]] of $R$. > > Each elementary operation itself has a corresponding [[matrix]]; left-multiplication acts on the rows of $P$ while multiplication on the right acts on the columns of $P$. These are: > > **1.** A [[permutation matrix]] enacting a transposition, i.e., of the form $\begin{bmatrix} > \boldsymbol e_{1} \cdots \boldsymbol e_{j} \cdots \boldsymbol e_{i} \cdots \boldsymbol e_{n} (\text{or } \boldsymbol e_{m}) > \end{bmatrix}$ > if rows/column $i,j$ of $P$ are to be swapped. > > **2.** Taking the [[identity matrix]] $\begin{bmatrix} > 1 & 0 & \cdots & 0 \\ > 0 & 1 & \cdots & 0 \\ > \vdots & \vdots & \ddots & \vdots \\ > 0 & 0 & \cdots & 1 > \end{bmatrix}$ > and replacing entry $ij$, $i \neq j$, with $c \in R$ yields a matrix that right-acts (resp. left-acts) on $P$ by adding a $c$-multiple of the $i^{th}$ column (resp. row) to the $j^{th}$ column (resp. row). > > > **3.** For $u$ a [[unit]], take the [[identity matrix]] $\begin{bmatrix} > 1 & 0 & \cdots & 0 \\ > 0 & 1 & \cdots & 0 \\ > \vdots & \vdots & \ddots & \vdots \\ > 0 & 0 & \cdots & 1 > \end{bmatrix}$ > and replace the $i^{th}$ diagonal element with $u$ to yield a [[matrix]] whose right-multiplication will scale the $i^{th}$ column by $u$ and left-multiplication will scale the $i^{th}$ row by $u$. > > > Each matrix is clearly invertible, e.g. because the corresponding [[linear operator|linear operators]] are. Indeed, such reasoning also implies that the inverse of an elementary matrix is again elementary. ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```