Description
The Jordan normal form has many important applications in algebra, often not covered in a first course in Linear Algebra. The Weyr canonical form of a matrix is a little known cousin of the Jordan canonical form. It turns out that the Weyr form is dual to the Jordan normal form in a certain sense. For example, consider the following two partitions of \(8\):
These diagrams illustrate that
\(8=4+3+1\) and \(8=3+2+2+1\), respectively, and one is obtained
from the other by turning columns into rows. While the Jordan form
of certain matrices of size 8 would have block sizes given by the
first partition, the Weyr form of the same matrix would have block
sizes given by the second partition.
While the Jordan form is closer to being diagonal, the Weyr form has several other advantages which make it superior in various problems in algebra. For example, there are recent results in various areas of algebra (including representation theory) where solutions to problems were obtained using the Weyr form, but where it is not known whether the Jordan form can be used.
After introducing the Weyr form and its basic properties, we will look at all or some of the following applications, each of which leads into a new branch of algebra:
- Centralisers of matrices (i.e., the ring of all matrices commuting with a given matrix). Centralisers are vector spaces and are fundamental for many things in group and representation theory. A classical formula of Frobenius says that if \(A\) is a nilpotent matrix (i.e., the only eigenvalue is \(0\)), then the dimension of its centraliser is \[m_1 + 3m_2 + 5m_3 +\cdots + (2s-1)m_s,\] where \((m_1, m_2, ..., m_s)\) are the sizes of the Jordan blocks of \(A\). This formula has a much simpler proof when using the Weyr form.
- The Gerstenhaber theorem, namely that if \(A\) and \(B\) are two commuting \(n\times n\) matrices over a field \(F\), then the ring \(F[A,B]\) generated by the two matrices over \(F\) has dimension at most \(n\) (as a vector space over \(F\)). A similar statement (much easier to prove; try it!) is that \(F[A]\) has dimension at most \(n\). On the other hand, if we use four matrices, then there are examples where the dimension is more than \(n\). For three commuting matrices, it is an intriguing open problem whether the ring has dimension at most \(n\). So far, no proof or counter-example has ever been found. The Weyr form gives an approach to this problem.There are also recent attempts to search for counter-examples to the Gerstenhaber problem (see reference 2 below). A study of this, and related computational techniques in algebra, can be an area of further study in the project.
- Basic notions of algebraic geometry. We will look at various proofs, which require the notions of dimension of an algebraic variety (a variety is a geometric space given by solutions to systems of polynomial equations), the Zariski topology and corresponding irreducibility, density etc. We will study these notions with the aim of understanding the arguments in the applications to Gerstenhaber's theorem and the following:
- Guralnick's theorem for \(\mathcal{C}(3,n)\). Let \(\mathcal{C}(k,n)\) denote the set of \(k\)-tuples of commuting \(n\times n\) matrices over an algebraically closed field \(F\). This is an algebraic variety. It has been known for some time that it is irreducible for \(k=1,2\) and for any \(k\) when \(n<4\), but that it is reducible for all \(k\geq 4\) when \(n\geq 4\). The only open case is therefore \(k=3\), that is, the irreducibility of the variety of commuting triples of matrices. For \(n\geq 8\) it has been checked (with increasingly complicated arguments) that \(\mathcal{C}(k,n)\) is irreducible. In 1992, Guralnick proved the striking result that \(\mathcal{C}(k,n)\) is reducible for \(n\geq 32\)! (that's 32 with an exclamation, not 32 factorial). Later refinements have narrowed this down to \(n\geq 30\), but we still don't know for which \(n\) \(\mathcal{C}(k,n)\) goes from being irreducible to reducible. We will study a version of Guralnick's elegant argument, but simplified using Weyr matrices.
Prerequisites
Algebra II.Resources
- K. C. O'Meara, J. Clark, C. I. Vinsonhaler, Advanced Topics in Linear Algebra.
- Holbrook and O'Meara, Some thoughts on Gerstenhaber's
theorem, link.