The number of congruence classes in \(M_ n(\mathbb{F}_ q)\)
type: Item
instance of: scholarly article
title: The number of congruence classes in \(M_ n(\mathbb{F}_ q)\)
zbMATH Open document ID: 0819.15012
author: William C. Waterhouse
published in: Finite Fields and their Applications
publication date: 1995-02-09T00:00:00Z
review text: The author calls two matrices \(A\) and \(B \in M_ n (K)\) congruent, when \(B = PAP^ t\) for some \(P \in GL_ n (K)\). The number of congruence classes is equal to the number of bilinear functionals on \(K^ n\). It is shown, that in the case of \(K = \mathbb{F}_ q\), this number is the coefficient of \(t^ n\) in \(\prod_{k \geq 1} (1 + t^ k)^ e (1 - qt^{2k}) (1 - t^ k)^{-1}\), where \(e = 1\) for \(q\) even and \(e = 2\) for \(q\) odd. For \(q\) even the number of classes where \(B^{-1} B^ t\) is unipotent is equal to the number of partitions of \(n\).
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 720502
zbMATH Keywords: number of congruence classes
zbMATH Keywords: number of bilinear functionals
reviewed by: Witold Więsław
MaRDI profile type: Publication
full work available at URL: ffta.1995.1004
OpenAlex ID: W2082117433
DOI: 10.1006/FFTA.1995.1004
description: scientific article; zbMATH DE number 720502