Definition ELEM Elementary Matrices
- For $i\neq j$, $\elemswap{i}{j}$ is the square matrix of size $n$ with
\begin{equation*}
\matrixentry{\elemswap{i}{j}}{k\ell}=
\begin{cases}
0 & k\neq i, k\neq j, \ell\neq k\\
1 & k\neq i, k\neq j, \ell=k\\
0 & k=i, \ell\neq j\\
1 & k=i, \ell=j\\
0 & k=j, \ell\neq i\\
1 & k=j, \ell=i
\end{cases}
\end{equation*}
- For $\alpha\neq 0$, $\elemmult{\alpha}{i}$ is the square matrix of size $n$ with
\begin{equation*}
\matrixentry{\elemmult{\alpha}{i}}{k\ell}=
\begin{cases}
0 & \ell\neq k\\
1 & k\neq i, \ell=k\\
\alpha & k=i, \ell=i
\end{cases}
\end{equation*}
- For $i\neq j$, $\elemadd{\alpha}{i}{j}$ is the square matrix of size $n$ with
\begin{equation*}
\matrixentry{\elemadd{\alpha}{i}{j}}{k\ell}=
\begin{cases}
0 & k\neq j, \ell\neq k\\
1 & k\neq j, \ell=k\\
0 & k=j, \ell\neq i, \ell\neq j\\
1 & k=j, \ell=j\\
\alpha & k=j, \ell=i\\
\end{cases}
\end{equation*}