# MA266 Multilinear Algebra

**Lecturer: **Christian BĂ¶hning

**Term(s):** Term 2

**Status for Mathematics students:** Core for MMath G103, Optional Core for BSc G100

**Commitment:** 30 one-hour lectures plus assignments

**Assessment:** 85% by 2-hour examination, 15% coursework

**Formal registration prerequisites: **None

**Assumed knowledge: **Knowledge of vector spaces and matrices from MA150 Algebra 2 or MA149 Linear Algebra or MA148 Vectors and Matrices. In particular, understanding change of basis matrices, eigenvalues and eigenvectors, elementary row and column operations and diagonalisation of matrices

**Useful background: **Group theory from MA151 Algebra 1 or MA267 Groups and Rings especially abelian groups

**Synergies:** The following modules go well with this module:

**Leads to: **The following modules have this module listed as **assumed knowledge** or **useful background:**

- MA3E1 Groups and Representations
- MA3H6 Algebraic Topology
- MA3J9 Historical Challenges in Mathematics
- MA3G6 Commutative Algebra
- MA3A6 Algebraic Number Theory
- MA377 Rings and Modules
- MA3F1 Introduction to Topology
- MA3K4 Introduction to Group Theory
- MA398 Matrix Analysis and Algorithms
- MA4C0 Differential Geometry
- MA453 Lie Algebras
- MA4H4 Geometric Group Theory
- MA4H0 Applied Dynamical Systems
- MA473 Reflection Groups

**Aims**: By the end of the module students should be familiar with the Jordan canonical form and some of its applications; have working knowledge of bilinear, quadratic and Hermitian forms and related theory; command the basic concepts of multilinear algebra in vector spaces and be comfortable to use arguments involving duals.

**Content**: This is a second linear algebra module. Its contents can be divided into three major groups.

The first main topic is the Jordan canonical form and related results. Abstractly, this solves the classification problem for pairs (V, T) where V is a finite dimensional vector space over the complex numbers (or any other algebraically closed field) and T a linear self-map of V, up to the equivalence relation induced by bijective linear self-maps of V; more concretely, we classify n by n complex matrices A up to conjugation by invertible matrices P, i.e., the operation A -> P^{-1}AP.

Secondly, we treat bilinear, sesquilinear and quadratic forms on finite dimensional (real and complex) vector spaces. These structures are ubiquitous and fundamental in mathematics and many parts of the sciences. For example, the standard scalar product in R^n is an example. In passing we mention that the description of amplitudes, probabilities and expectation values in quantum theory places such structures at the very heart of how nature works at the smallest levels. We will cover orthonormal basis, Gram-Schmidt process, diagonalisation, singular value decomposition, hermitian forms and normal matrices, among other things.

The third part is concerned with a thorough discussion of the very useful concept of duality (dual vector spaces, dual linear maps, dual bases etc.) and its applications, and after that tensor, exterior and symmetric algebras and their basic properties.

**Objectives**:

**Books: **

P M Cohn, *Algebra*, Vol. 1, Wiley, 1982

I N Herstein, *Topics in Algebra*, Wiley, 1975

Jorg Liesen and Volker Mehrmann, *Linear Algebra*, Springer, 2015

Peter Petersen, *Linear Algebra*, Springer, 2012

F. Gantmacher, *The Theory of Matrices*, American Mathematical Society, 2001

Peter Lax, *Linear Algebra and Its Applications*, 2nd Edition, Wiley, 2007