Geometrical meaning of $ \begin{pmatrix} a\\ b \end{pmatrix} \mapsto \begin{pmatrix} a&-b\\ b&a \end{pmatrix}$ 
Geometrical meaning of $\tau: \mathbb{R^2} \rightarrow \mathbb{R^{2,2}}, \begin{pmatrix} a\\ b \end{pmatrix} \mapsto \begin{pmatrix} a&-b\\ b&a \end{pmatrix}$

I have troubles of reading matrices and interpreting them. How can you interpret them? Here my best guess is, that a vector is mapped into a higherdimensional space. In the image, do I have to look at the columns to see the vectors, does that even make sense or any of the images below? 
EDIT:


You drew your vector wrong. The green vector should have x-coordinate -b and y-coordinate a. If you remember from algebra class that if a line has slope m, then a perpendicular line has slope -1/m. Therefore it is a mapping of a vector to itself and a perpendicular vector.
