Spectra and Tensors: Topics in Linear Algebra
Course Description
Summary
This intermediate/advanced linear algebra class will focus mostly on two main topics: spectra, and tensors. The theory of matrix spectra, (eigenvalues and eigenvectors), is arguably one of the most useful ideas in all of mathematics. We will discuss some major applications, such as input-output models in economics, the Page Rank algorithm, JPEG compression, the Fast Fourier Transform, heat and wave equations, and principal component analysis in statistics and machine learning. Spectra also underlie the fundamental structure of physical reality, in their central role in quantum mechanics. We will also discuss interesting theoretical questions: what is the "best" representation of a matrix? What distinguishes particular families of matrices, such as rotations? This will lead into such topics as introductory Lie groups.
The other topic of the course will be tensors, which are a generalization of vectors and matrices. The approach here will be fairly classical and concrete, with an aim of having students be able to understand the application of tensors to geometry, physics, and computer science.
Along the way, the class will also introduce some necessary foundations in linear algebra, building on a first linear algebra course. This will include some theory of determinants.
The approach will be a mixture of modern and historically oriented, depending on what provides clearest motivation for the concepts and the problems.
Students may take this class at different levels of mathematical intensity: it can either be an intermediate or advanced class, depending on how much of the mathematical detail you get into. What level an individual student completes will be reflected in the narrative evaluation. The final evaluation will be relative to the student goals. For example, a student can get an "A" if they choose to take this as an intermediate class, doing only the core ideas, and doing a good job with those; the evaluation would identify that the student took it as an intermediate class.
The prerequisite for the class is MAT 2482 Linear Algebra: An Introduction, or a similar background.
Learning Outcomes
- to understand the fundamentals of eigenvalues, eigenvectors, and matrix decompositions
- to understand the fundamentals of tensors
- to understand fundamental theorems in the subject (e.g. Principal Axis Theorem, Cayley-Hamilton Theorem)
- to be familiar with at least some of the main applications of the above (e.g. FFT, principal component analysis)
- to become more adept with linear algebra tools, techniques, and concepts, beyond the fundamentals of a first course
Prerequisites
The prerequisite for the class is MAT 2482 Linear Algebra: An Introduction, or a similar background.
Please contact the faculty member : amcintyre@bennington.edu