Elementary matrix

Definition
Let $$R$$ be a unital ring and $$n$$ be a natural number. An elementary matrix is an element of the general linear group $$GL_n(R)$$ that is of one of these three types:


 * Elementary matrix of the first kind, also called a shear matrix: This has $$1$$s on the diagonal and at most one nonzero off-diagonal entry.
 * Elementary matrix of the second kind, also called a transposition matrix: This is the matrix for a transposition permutation.
 * Elementary matrix of the third kind, which is a diagonal matrix with $$1$$s on all diagonal entries except at most one place, and that entry also needs to be nonzero.

Often, the term elementary matrix is used exclusively for an elementary matrix of the first kind.

Facts

 * Elementary matrices of the first kind generate the special linear group over a field
 * Elementary matrices of the first and third kind generate the general linear group over a field