# Describing the action of this Matrix

linear algebralinear-transformationsmatricesvisualization

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:

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".