There is an important link with geometry as follows. The group $\text{GL}_2^+(\Bbb R)$ of matrices with positive determinant acts on the complex upper halfplane $ \cal H=\{z=x+iy\in\Bbb C\text{ such that }y>0\} $ by linear fractional transformations $ \left(\begin{array}{cc}a&b \\ c&d \end{array}\right)\cdot z=\frac{az+b}{cz+d} $ Parenthetically, these transformations are isometries when $\cal H$ is given the metric that makes it a model of the hyperbolic plane. Then the matrices $ M=\left(\begin{array}{cc}a&b \\ -b&a \end{array}\right) $ are exactly those that stabilize $i$, i.e. $M\cdot i=i$, and form a subgroup isomorphic to the multiplicative group $\Bbb C^\times$. It turns out that the other subgrous of $\text{GL}_2(\Bbb R)$ isomorphic to $\Bbb C^\times$ are precisely the stabilizers of the various points of $\cal H$.
Finally, the fact all these subgroups are conjugated is the "translation" of the fact that the described action is transitive, as $ \left(\begin{array}{cc}y&x \\ 0&1 \end{array}\right)\cdot i=x+iy. $