Topics in Linear Algebra
This course will meet Mondays and Wednesdays, 12:00-1:20, in Lang 312. It will count towards the algebra breadth requirement. I plan to run the lectures according to a kind of Socratic method: I will ask you questions leading to the results I would like to cover, and you will work together to solve them, with hints from me as needed. In each class, one of you will act as “scribe”, standing at the board to record the communal progress. The rest of you will brainstorm. The role of scribe will rotate through the class. There will be no homework or exams: grades will be based on attendance and participation.
The general subject will be linear algebra. The specific topics are flexible, and I am open to suggestions: let me know if there is something in particular you would like to learn. Here are some possible directions:
What really is the determinant, why is it non-zero precisely on the invertible matrices, and why is it multiplicative? Everyone thinks they already know these things and feels bored at the idea of going back over them. But I am not so sure everyone really does! So we could compromise by learning about the Pfaffian. This is a polynomial function of degree N on the space of skew-symmetric 2N by 2N matrices which is a square root of the determinant. Proving stuff about it is actually just like proving stuff about the determinant, only less boring, because square roots are always cool, and this one has a particularly cool name.
We will definitely discuss the fundamental fact about matrices: associated to any N by N matrix is the decomposition of C^N into the direct sum of its generalized eigenspaces. Maybe you have seen this. Maybe you have even learned about the Jordan normal form. The Jordan form is pretty hard, but we can understand the generalized eigenspace decomposition much more easily.
And once we understand the generalized eigenspace decomposition, we can understand an important corollary: any (square) matrix may be written as the sum of a nilpotent matrix and a diagonalizable matrix, which commute with each other. Maybe you don’t find that statement so impressive, but it becomes more impressive when you realize that the nilpotent matrix and the diagonalizable matrix are uniquely determined by the original matrix. This is the Jordan decomposition.
And another corollary of the generalized eigenspace decomposition: given any matrix, we can describe its commutant, the algebra of all matrices which commute with it.
And another: we can describe the joint generalized eigenspaces of an arbitrary pair of commuting matrices. Or any number of commuting matrices. (But we can’t classify pairs of commuting matrices, because that’s impossible.)
The generalized eigenspace decomposition is a (large) step towards understanding the conjugacy classes of the group GL(N, C) of all invertible matrices. We could also learn about the conjugacy classes of the groups O(N, R) (and SO(N, R)) of real matrices which preserve length (and orientation). For example, why is it that if you find a kugel fountain, rotate it by some angle about some axis, and then rotate it by another angle around another axis, the two motions combined amount to a single rotation by some third angle around some third axis? And why is this false in four dimensions?
And how do you find that third axis? This question could lead us to study the conjugacy classes of the sphere in four-dimensional space, which, unlike the sphere in three-dimensional space, is a group.
That last topic is really the quaternions. If there is interest in linear algebra over finite fields, we could use the conjugacy classes of the quaternions over the field of 7 elements to analyze the simple group of 168 elements. And while we are there, we might also look at the same group through the lens of 3 by 3 matrices over the field of 2 elements.
In a completely different direction, if there is interest in Galois theory, we could see how to use linear algebra over the field defined by the square root of -1 together with the square root of the quantity 2 plus the square root of 2 to construct the regular 17-gon with a straightedge and compass.