Transvection

Definition

Let $R$ be a commutative ring, let $n \ge 2$, let $i \ne j$ be indices in $\{1, \dots, n\}$ and let $c \in R$. Let $E_{ij}$ be the $n \times n$ matrix with entry $1$ in position $(i, j)$ and $0$ elsewhere. The transvection matrix is $$T_{ij}(c) = I_n + c E_{ij}.$$

Remarks

The product $T_{ij}(c) A$ adds $c$ times row $j$ of $A$ to row $i$; this is an elementary matrix of the row-addition type. Its determinant is $1$, and its inverse is $T_{ij}(-c)$. Over a field, the transvections generate the special linear group. In the plane, $\begin{pmatrix} 1 & c \\ 0 & 1 \end{pmatrix}$ is a horizontal shear.

Explanations

L1 human

A transvection slides each point parallel to one axis, by an amount that depends on another coordinate. A deck of cards that we push sideways is a picture of a shear. Areas and volumes do not change, which matches the determinant $1$.