elementary matrixElementaryMatrix2008-10-21 00:21:442008-10-22 11:15:11Definitionmatrixelementary operationelementary column operationelementary row operationbasic diagonal matrixtransposition matrixrow replacement matrix\usepackage{amssymb,amscd}
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\newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}}\subsubsection*{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 \emph{elementary row operation} if it takes a matrix $M\in \mathbb{M}$, and does one of the following:
\begin{enumerate}
\item interchanges of two rows of $M$,
\item multiply a row of $M$ by a non-zero element of $R$,
\item add a (constant) multiple of a row of $M$ to another row of $M$.
\end{enumerate}
An \emph{elementary column operation} is defined similarly. An operation on $\mathbb{M}$ is an \emph{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
\begin{enumerate}
\item $\begin{pmatrix} a & b \\ e & f \\ c & d \end{pmatrix}$ is obtained by interchanging rows 2 and 3 of $M$,
\item $\begin{pmatrix} a & b \\ rc & rd \\ e & f \end{pmatrix}$ is obtained by multiplying $r\ne 0$ to the second row of $M$,
\item $\begin{pmatrix} a & b \\ c & d \\ sa+e & sb+f \end{pmatrix}$ is obtained by adding to row 1 multiplied by $s$ to row 3 of $M$.
\end{enumerate}
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 $k$.
\subsubsection*{Elementary Matrices}
Now, assume $R$ has $1$. An $n\times n$ \emph{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:
\begin{enumerate}
\item \emph{transposition matrix} $T_{ij}$: an matrix obtained from $I_n$ with rows $i$ and $j$ switched,
\item \emph{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 $R$
\item \emph{row replacement matrix} $E_{ij}(s)$: $I_n + s U_{ij}$, where $s\in R$ and $U_{ij}$ is a matrix unit with $i\ne j$.
\end{enumerate}
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:
\begin{itemize}
\item $T_{ij}=T_{ji}$, and $T_{ij}^2=I_n$.
\item $D_i(r)D_i(r^{-1})=I_n$, provided that $r^{-1}$ exists.
\item $E_{ij}(s) E_{ij}(-s) = I_n$.
\item $\det(T_{ij})=-1$, $\det(D_i(r))=r$, and $\det(E_{ij}(s))=1$.
\item If $A$ is an $m\times n$ matrix, then $$E_c^k(A)=\lbrace AE \mid E \in \mathbb{E}^k(n) \rbrace \qquad \mbox{and} \qquad E_r^k(A)=\lbrace EA \mid E \in \mathbb{E}^k(m) \rbrace.$$
\item 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.
\end{itemize}
\textbf{Remark}. 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.