# Visualising the Real Orthogonal Group

Posted by Jason Polak on 09. June 2013 · Write a comment · Categories: group-theory · Tags: ,

In the post Can You See in Four Dimensions?, we saw some ways of visualising functions plotted in four-dimensional ‘space’ in various ways. Of course, we used colour and time for two dimensions because it is a bit difficult to plot in four actual spatial dimensions!

Here is another example: the orthogonal group $\mathrm{O}_2(\mathbb{R})$ over the real numbers $\mathbb{R}$. This is the group of all matrices $A\in \mathrm{GL}_2(\mathbb{R})$ such that $AA^t = I_2$, where $I_2$ is the $2\times 2$ identity. Let

$A = \begin{pmatrix}a & b\\ c & d\end{pmatrix}$

be such a matrix. Then $a = \pm\sqrt{1 – b^2}$ and $c = \pm\sqrt{1 – d^2}$. Finally, $ac + bd =0$. As long as these conditions are satisfied, then $A$ will indeed be in the orthogonal group.
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