A Generalised Jordan Normal Form and Its Computation Over Finite Fields

Abstract

The question of matrix similarity is a classical one in linear algebra. For a field $\mathbb{F}$ and some positive integer $n \in \mathbb{N}$, one may consider the following problems: 1. Given two matrices $A, B \in \mathrm{GL}(n, \mathbb{F})$, determine whether they are similar or not. 2. If they are similar, compute a conjugating matrix $X \in \mathrm{GL}(n, \mathbb{F})$. 3. List a representative for each conjugacy class of $\mathrm{GL}(n, \mathbb{F})$. They can be readily solved by using normal forms. The most commonly studied forms are the rational canonical form (also known as the Frobenius normal form) and the Jordan normal form. The Jordan form, however, is traditionally defined only over algebraically closed fields such as $\mathbb{C}$. In this thesis, we aim to extend the notion of the Jordan normal form to arbitrary fields. Moreover, we provide practical algorithms for computing this generalized Jordan form, which we have implemented in GAP for finite fields. The construction of the Jordan normal form relies on analyzing the action of a matrix $A \in \mathbb{F}^{n\times n}$ on the vector space $V = \mathbb{F}^n$. By decomposing $V$ into $A$-invariant subspaces, one obtains, in a sense, a corresponding decomposition of $A$ itself. The proofs in this thesis are expressed in terms of matrices, rather than modules, to reflect the computational approach used in practice.