Elementary matrices/Definition

Elementary matrix

Let ${}K$ be a field. We denote by ${}B_{ij}$ the ${}n\times n$-matrix, with entry ${}1$ at the position ${}(i,j)$ and entry ${}0$ everywhere else. Then the following matrices are called elementary matrices.

1. ${}V_{ij}:=E_{n}-B_{ii}-B_{jj}+B_{ij}+B_{ji}$.
2. ${}S_{k}(s):=E_{n}+(s-1)B_{kk}{\text{ for }}s\neq 0$
3. ${}A_{ij}(a):=E_{n}+aB_{ij}{\text{ for }}i\neq j{\text{ and }}a\in K$