Describing the action of this Matrix?

Let $A = \begin{bmatrix} 1 & 1 \\ 0 &-1 \end{bmatrix}$

Then $A \begin{bmatrix} 1 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} $ and $A \begin{bmatrix} 0 \\ 1 \end{bmatrix} = \begin{bmatrix} -1 \\ 1 \end{bmatrix} $

My first thought is that this is a reflection, or some combination of reflections, but I can't quite concisely describe it.

For example for $\chi_1 = \{x_1 + x_2 = 1\}$ we find $A : \chi_1 \to \{x=1\}$ with the action visualized below:

And for $\chi_2 = \{x_1 - x_2 = 1\}$ we find $A : \chi_1 \to \{2x_2 + x_1 = 1\}$ with the action visualized below

We also see A acting on the unit balls:

Solution 1:

This kind of transformation is sometimes called an oblique reflection. For a point $(x_1,x_2)$, we can describe the corresponding output as follows.

Draw the line through $(x_1,x_2)$ parallel to the vector $(-1,2)$ (the eigenvector of $A$ associated with $-1$). The output is the point on this line whose distance from the $x_1$-axis (the line parallel to $(1,0)$, which is the eigenvector of $A$ associated with $1$) is the same as the distance of $(x_1,x_2)$ from the $x_1$-axis.

What makes this reflection "oblique" is the fact that $(-1,2)$, the "direction of reflection", is not orthogonal to $(1,0)$, the "axis of reflection".

Solution 2:

The matrix is the product $$\begin{pmatrix} 1 &0 \\0 &-1\end{pmatrix}\cdot \begin{pmatrix} 1 &1\\0 &1\end{pmatrix}.$$

So it is a composition of a shearing and a reflection. But it is not a product of reflections (and rotations) because it does not preserve Euclidean lengths:

$$\begin{pmatrix} 1 &1 \\0 & -1 \end{pmatrix}\begin{pmatrix} 0\\1\end{pmatrix}=\begin{pmatrix} 1\\ -1\end{pmatrix}.$$

Here you can find more info about linear transformations of $\Bbb R^2$.