Techniques in Linear Algebra workshop

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A two day workshop to on topics in Linear algebra.

Linear algebra has numerous applications throughout mathematics. This workshop will discuss two fundamental classification results in linear algebra: the Jordan Canonical form for linear operators, and the Hasse-Minkowski theorem for quadratic forms. Particular emphasis will be placed on the ideas and techniques used in the proofs. The workshop is suitable for advanced undergraduate students, graduate students and anyone else with an interest in reviewing these classical algebraic topics. The only pre-requisite is a thorough understanding of elementary linear algebra: conceptual understanding of diagonalisation of a matrix should suffice.

The first day will begin with a quick review of eigenvalues and eigenvectors before giving the complete analysis of a linear operator in terms of generalised eigenvalues. We will explain the significance of the Cayley—Hamilton theorem (and why, in our opinion, no-one states the result appropriately). We’ll conclude with the Jordan Canonical Form. Our approach will be mostly elementary, though we will use the word 'module' on occasion. 

On the second day, we will discuss the classification of quadratic forms, with particular attention to the fields Q, R, C. This generalises the theory of inner product spaces (which is often introduced but rarely mastered at undergraduate level). We will discuss the polarisation of forms, the classification using Hilbert symbols and state the Local—Global principal. We focus on concrete computations: knowledge of p-adic numbers will not be required! As an application, we will give a complete and self-contained proof of the Bruck—Ryser—Chowla theorem, ruling out the existence of certain projective planes.

 To help with catering and numbers, please register if you plan on attending; there is no registration fee.

Parking OUTSIDE of P&D areas on campus is free.

For any help or extra details, please email:

patrick.browne@tus.ie