Visualising the Real Orthogonal Group


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.

If we specify $ a$, then there are two choices for $ b$, and for each choice of $ c$, there are two choices for the row $ (c,d)$. We could specify $ a$ as a time, and then for each choice of $ b$, we can plot a point at each possibility for $ (c,d)$. Since $ b\in [-1,1]$, we can choose a colour by generating a rainbow spectrum from red to violet over the interval $ [-1,1]$.

This is what we get, where the value of $ a$ is displayed at the top:


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