Groups Lecture 19

in which we act with matrices on vectors and on matrices, and meet the orthogonal group.

Today we looked at two different actions of \mathrm{GL}_n(\mathbb{C}) (\mathrm{GL}_n(\mathbb{R}) works similarly). The group \mathrm{GL}_n(\mathbb{C}) acts on \mathbb{C}^n faithfully with two orbits, by viewing the matrices as linear maps, i.e. “applying the function to a vector”.  As they are all invertible linear maps, 0 forms a singleton orbit and all the other vectors form an orbit together. You were challenged to find out what orbits there are under the action of \mathrm{SL}_n(\mathbb{C}) on \mathbb{C}^n.

Another (possibly more interesting) way that \mathrm{GL}_n(\mathbb{C}) acts is by conjugation on the set M_{n\times n}(\mathbb{C}) of n\times n matrices. 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. 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.

At the end of the lecture we introduced the orthogonal group \mathrm{O}_n=\mathrm{O}_n(\mathbb{R}), the group of those matrices satisfying A^TA=I. We showed that orthogonal matrices are isometries. We have started to investigate these more closely, especially for n=2 and 3: SO_2 consists of all rotations of \mathbb{R}^2 around 0. More on this next time.

Understanding today’s lecture

The material today has a lot of intersection with and relevance for Vectors and Matrices. You will benefit from looking at both sources together, seeing what results from one course you use/need in the other, how they might give different viewpoints or proofs for the same thing, how they build up a better picture for you of understanding matrices as linear maps.

Preparing for Lecture 20

Next time will be all about rotations and reflections in 2 and 3 dimensions. Brush up on a bit of trigonometry, in particular how cosine and sine of an angle relate to the sides of a right-angled triangle. You’ll need some matrix multiplication, and eigenvalues also come into it quite a lot. More specifically, we will prove that matrices in \mathrm{SO}_2 and \mathrm{SO}_3 are rotations, that any matrix in \mathrm{O}_2 is a rotation or a reflection in a line through 0, and that any matrix in \mathrm{O}_3 is a product of at most 3 reflections (in planes through 0). You can read ahead in Beardon Section 11.2 Orthogonal maps, and result 11.3.3, but I will do different proofs than Beardon.

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. The orbit of the Rosetta spacecraft whose lander just landed on a comet is an example of a very carefully chosen orbit of \mathbb{R}. 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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