Groups Lecture 19

in which meet the orthogonal group and we rotate and reflect in two and three dimensions.

In the beginning of the lecture we reminded ourselves how \mathrm{GL}_n(\mathbb{C}) acts by conjugation on the set M_{n\times n}(\mathbb{C}) of n\times n matrices: PAP^{-1}. We can think of this as base change: two conjugate matrices represent the same linear map \mathbb{C}^n\to \mathbb{C}^n with respect to different bases, and P is a base change matrix. Then we get some very nice representatives for each conjugacy class (i.e. orbit), given by the Jordan Normal Form. You have seen this for 2\times 2 matrices in Vectors and Matrices, and will see it for general n in second year Linear Algebra.

We then introduced the orthogonal group \mathrm{O}_n=\mathrm{O}_n(\mathbb{R}), the group of those matrices satisfying A^TA=I, together with its partner the special orthogonal group \mathrm{SO}_n=\mathrm{SO}_n(\mathbb{R}), which is the kernel of the determinant homomorphism. Orthogonal matrices have determinant \pm 1, so \mathrm{O}_n/\mathrm{SO}_n\cong C_2. We showed that orthogonal matrices are isometries, which means they preserve dot product and length of vectors.

For n=2 and 3 we investigate these orthogonal groups more closely: \mathrm{SO}_2 consist of all rotations of \mathbb{R}^2 around 0, and any matrix in \mathrm{O}_2 is a rotation or a reflection (in a line through 0). We made use of the fact that \mathrm{O}_2 is partitioned into the two cosets \mathrm{SO}_2 and \left(\begin{smallmatrix}1 & 0\\ 0& -1\end{smallmatrix}\right)\mathrm{SO}_2.

Going to three dimensions, every matrix in \mathrm{SO}_3 is a rotation around some axis, which we proved using some eigenvalue-knowledge from V+M and the two-dimensional situation. Next time we will conclude that any matrix in \mathrm{O}_3 is the product of at most three reflections, using cosets again. We really do need three for -I, which shows that we can’t quite think of \mathrm{O}_3 as “rotations and reflections”, at least not if we mean just reflections in planes.

Understanding today’s lecture

Today we mixed some general group theory results into geometric understanding of specific matrix groups. We used some geometric knowledge, like knowing what a rotation in two dimensions looks like, knowing things about eigenvalues and eigenvectors etc, with some algebraic knowledge, like that cosets partition a group. Personally I find this interplay very helpful to deepen understanding. If you find that you can think more easily in geometric terms, I recommend looking at Beardon “Algebra and Geometry” Section 11.2 and 11.3.3 to see how these results can be dealt with more geometrically.

Preparing for Lecture 20

We will be entering into the territory of Möbius maps next time. Come equipped with stamina for fraction manipulation (with lots of variables), and an open mind about infinity.

Going a little deeper

These matrix groups we are studying now have some special properties our other (mostly finite) groups don’t really have. First of all, they are infinite groups. But because \mathbb{R}^n and \mathbb{C}^n have distances and analysis and geometry on them, these matrix groups also have the structure of so-called differentiable manifolds. Essentially that means that locally they look like a linear space on which you can do calculus. We have nowhere near enough tools and knowledge yet to study these in first year, but it is important to know that these matrix groups are more than just groups. They are what is called Lie groups. Lie groups are ubiquitous in Physics, and they are the “structure groups” of many important physical theories. But don’t get the impression from our first example that orbits of Lie groups are always “easy”. Action of the reals could have terribly complicated orbits.  Apparently people also make patents of orbits!

You can look up Lie groups if you are interested, but you’ll only get to study them properly in courses in third or fourth year. These references might be a good start: Carter, Segal, McDonald (Segal’s Chapter) “Lectures on Lie groups and Lie algebras” (LMS), and Adams, “Lectures on Lie groups” (U. Chicago Press). If you’re very interested, I have a few exercises that were given to me by an expert, which could set you going (but you need some analysis).

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