The general linear group

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 13 August 2013

An $n \times n$ matrix $A$ is invertible if there is a matrix $A^{- 1}$ such that $A A^{- 1} = A^{- 1} A = I_{n} .$
${G L}_{n} (R) = {n \times n invertible matrices with entries in R} .$

${G L}_{n} (R)$ is a group with identity $I_{n} .$

Let $R$ be a ring and let $R^{*}$ be the group of units in $R .$

The matrices $e_{i j}$ which contain a 1 in the $i^{th}$ row and the $j^{th}$ column and zeros in all other entries are called matrix units.
The elementary matrices are $\begin{matrix} x_{i j} (t) = (\begin{matrix} 1 \\ ⋱ & t \\ ⋱ \\ ⋱ \\ 1 \end{matrix}) & for 1 \leq i, j \leq n, i \neq j, t \in R, \\ s_{i j} = (\begin{matrix} 1 & 0 \\ ⋱ \\ 1 \\ 0 & 1 \\ 1 \\ ⋱ \\ 1 \\ 1 & 0 \\ 1 \\ ⋱ \\ 0 & 1 \end{matrix}), & 1 \leq i, j \leq n, i \neq j, \\ h_{i} (t) = (\begin{matrix} 1 \\ ⋱ \\ 1 \\ t \\ 1 \\ ⋱ \\ 1 \end{matrix}) & 1 \leq i \leq n, t \in R^{*} . \end{matrix}$

Let $A$ be an $n \times n$ matrix.

\begin{matrix} x_{i j} (t) A & is the same as A except that & t \cdot (j^{th} row of A) is added to the i^{th} row of A . \\ A x_{i j} (t) & is the same as A except that & t \cdot (j^{th} column of A) is added to the i^{th} column of A . \\ s_{i j} A & is the same as A except that & the j^{th} row of A and the i^{th} row of A are switched. \\ A s_{i j} & is the same as A except that & the j^{th} column of A and the i^{th} column of A are switched. \\ h_{i} (t) A & is the same as A except that & the i^{th} row of A is multiplied by t . \\ A h_{i} (t) & is the same as A except that & the i^{th} column of A is multiplied by t . \end{matrix}

(Smith normal form) Let $R$ be a PID. Any matrix $A \in M_{m \times n} (R)$ can be written in the form

A = P D Q, where D = diag (d_{1}, \dots, d_{r}, 0, \dots, 0),

where $P \in {G L}_{m} (R),$ $Q \in {G L}_{n} (R)$ and $d_{1}, \dots, d_{r} \in R$ with $d_{i} | d_{i + 1}$ for $1 \leq i \leq r - 1 .$