Canonical forms of two by two by two matrices
type: Item
instance of: scholarly article
title: Canonical forms of two by two by two matrices
zbMATH Open document ID: 0920.15012
author: Richard Ehrenborg
published in: Journal of Algebra
publication date: 1999-09-08T00:00:00Z
review text: The aim of this paper is to determine the canonical forms of two by two by two matrices. Firstly, an invariant \(\Gamma\) of two by two by two matrices is introduced. It is shown by a Gaussian elimination argument that any invariant of two by two by two matrices has the form \(s\cdot\Gamma^k\) where \(s\) is a constant and \(k\) is a nonnegative integer. Then 6 covariants are presented in order to separate the canonical forms. After this the author presents 7 different canonical forms for the two by two by two matrices. Among that, \(pqr+stu\) is a generic canonical form. Hence it is easy to prove certain relations between these covariants. In fact, the binary cubic may be viewed as symmetric two by two by two matrices. Thus there is a natural correspondence between the canonical forms and covariants of two by two by two matrices and binary cubics. Similarly, two by two by two matrices are also related to skew-symmetric tensors of step 3 in the exterior algebra Ext\((U)\), where \(\dim (U)=6\).
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 1270880
zbMATH Keywords: canonical forms
zbMATH Keywords: covariant
zbMATH Keywords: skew-symmetric tensor
zbMATH Keywords: binary cubic
zbMATH Keywords: two by two by two matrices
zbMATH Keywords: Gaussian elimination
MaRDI profile type: Publication
full work available at URL: jabr.1995.6650
OpenAlex ID: W2100821645
cites work: Apolarity and canonical forms for homogeneous polynomials
cites work: On apolarity and generic canonical forms
cites work: Q3792822
cites work: The football pool problem for 5 matches
cites work: The invariant theory of binary forms
cites work: Note on a theorem of Zaremba
DOI: 10.1006/JABR.1995.6650
description: scientific article; zbMATH DE number 1270880