Handbook of Linear Algebra, 2nd Edition

Chapter 31

Matrices over Finite Fields

J. D. Botha

University of South Africa

31.1 Counting Matrices with a Given Property

Let F denote a finite field with q elements, i.e., F = GF(q), where q = pk for some positive integer k and prime p. Since the number of n × m matrices over F is finite (|Fn×m| = qnm), a number of results have appeared which count matrices with various properties. The following is a selection of some of these results.

Definitions:

A matrix A ∈ Fn×n is orthogonal if AT A = In.

A matrix A ∈ Fn×n is nonderogatory if its minimum and characteristic polynomials are equal.

The Euler function φ(f(x)) for polynomials over F is defined as the number of monic polynomials over F which are of the same degree as f(x) and relatively prime ...