DescriptionThe quaternions were discovered on 16th October 1843 by Sir William Rowan Hamilton. He published his discovery in the Proceedings of the Royal Irish Academy less than a month later!The quaternions are a generalisation of the complex numbers where we allow three distinct square roots of minus 1, called i, j, k. These anti-commute, meaning that ij=-ji=k, jk=-kj=i, ki=-ik=j. A quaternion is a+bi+cj+dk where a, b, c, d are real numbers. We can form matrices of quaternions and then try to mimic constructions from linear algebra such as inverting a matrix, finding eigenvalues and eigenvectors, diagonalising a matrix. The fact that quaternions do not commute makes these constructions much harder. For example, in order to find eigenvalues of a 2x2 quaternionic matrix one needs to find the roots of the characteristic polynomial. In other words one needs to solve a quadratic equation. Think about how you might go about doing this without ever using commutativity! In this project you will first learn about the quaternions and then take some aspect of linear algebra and try to generalise it to linear algebra defined over the quaternions. Many of the papers in the subject were written over 50 years ago. You will need to read these papers and give an accessible account of what they contain.
The project will be supervised by John Parker. Mode of operation and evidence of learningThe mode of delivery will be through weekly meetings, either in the group or individually. We will usually meet as a group throughout the first term. In this phase of the project we will talk about some general features of the subject which will provide a foundation for your project. Then I expect to meet with you individually in the second term. In this phase you will specialise to a topic within the area. The evidence of learning will be based on how much of the material you understand and can explain to others.
NoteI plan to be away from Durham in the first half of Epiphany Term and so we will meet online then. Otherwise meetings will be in person.ResourcesWeb based resources about quaternions:Early papers on quaternionic linear algebra: More recent papers on quaternionic linear algebra: |