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:

## Best Answer

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".