Shear inverse transformation

Question

Find the inverse of the shear transformation, and describe it geometically.

Matrix transformation T $\begin{bmatrix}x\\y\end{bmatrix}$=$\begin{bmatrix}x+ay\\y\end{bmatrix}$, induced by matrix A = $\begin{bmatrix}1&a\\0&1\end{bmatrix}$.

Answer 1

It is clear that inverting:

$$\begin{cases}x'&=&x+ay\\ y'&=&y\end{cases} \ \iff \ \begin{cases}x&=&x'-ay'\\ y&=&y'\end{cases}$$

is transcripted into matrix notations:

$$\begin{bmatrix}x'\\y'\end{bmatrix}=\underbrace{\begin{bmatrix}1&a\\0&1\end{bmatrix}}_S\begin{bmatrix}x\\y\end{bmatrix} \ \iff \ \begin{bmatrix}x\\y\end{bmatrix}=\underbrace{\begin{bmatrix}1&-a\\0&1\end{bmatrix}}_{S^{-1}}\begin{bmatrix}x'\\y'\end{bmatrix}$$

$S^{-1}$ is thus obtained in a natural way...

More generally, it is interesting to see that the set of transvections with the general multiplication rule:

$$\underbrace{\begin{bmatrix}1&u\\0&1\end{bmatrix}}_{T_u}\underbrace{\begin{bmatrix}1&v\\0&1\end{bmatrix}}_{T_v}=\underbrace{\begin{bmatrix}1&u+v\\0&1\end{bmatrix}}_{T_{u+v}}$$

constitute a group isomorphic to $(\mathbb{R},+)$.

Fig. 1: The action of operators $T_u$ on a unit rectangle (colored in cyan). The neutral element corresponds to this unit rectangle. The multiplication law is illustrated here on $T_1 T_{2.5}=T_{3.5}$. Please note as well the visualization of the action of $T_{-1}$, inverse of $T_{1}$.