# Anders Johansson: Matrix Invariants

BSc thesis presentation

**Time: **
Thu 2020-06-04 10.30 - 11.30

**Lecturer: **
Anders Johansson

**Location: **
Zoom, meeting ID: 647 0784 5619

**Supervisor: **
Torbjörn Tambour

### Abstract

The set of invertible square matrices in the set of all square matrices is a group under matrix multiplication, and it acts on the set of all square matrices through conjugation. Invariant polynomial functions, or invariants, to this group action are polynomials in the elements of the matrix that are constant on the orbits. The main result of this thesis, is that all invariants can be expressed in terms of a few special invariants. These special invariants are the coefficients in the characteristic polynomial of the matrix that is subject to the group action. To do this we will look at the properties of permutations and linear transformations, their matrix representations as well as the characteristic polynomial of a matrix and its connection to these special invariants. We will also look at properties of polynomials, mainly symmetric polynomials and use the fundamental theorem of symmetric polynomials to find the invariants for different types of matrices.