Geometric matrix algebra
type: Item
instance of: scholarly article
title: Geometric matrix algebra
zbMATH Open document ID: 1149.15024
author: Garret Sobczyk
published in: Linear Algebra and its Applications
publication date: 2008-08-06T00:00:00Z
review text: Matrix multiplication was first introduced by Arthur Cayley in 1855 in agreement with the composition of linear transformations. The author explores an underlying geometric framework in which matrix multiplication naturally arises from the product of numbers in a geometric (Clifford) algebra. Consequently, all invariants of a linear operator become geometric invariants of the multivectors that they represent. Two different kinds of bases for matrices emerge, a spectral basis of idempotents and nilpotents, and a standard basis of scalars, vectors, bivectors, and higher order \(k\)-vectors. The Kronecker product of matrices naturally arises when considering the block structure of a matrix. Conformal geometry of \(\mathbb R^{3}\) is expressed in terms of the concept of an \(h\)-twistor, which is a generalization of a Penrose twistor.
reviewed by: Piwen Yang
Mathematics Subject Classification ID: 15A66
Mathematics Subject Classification ID: 15B33
Mathematics Subject Classification ID: 51F25
zbMATH DE Number: 5307056
zbMATH Keywords: Clifford algebra
zbMATH Keywords: conformal transformation
zbMATH Keywords: geometric algebra
zbMATH Keywords: horosphere
zbMATH Keywords: Kronecker product
zbMATH Keywords: Möbius transformation
zbMATH Keywords: quaternions
zbMATH Keywords: twistor
zbMATH Keywords: matrix multiplication
zbMATH Keywords: geometric invariants
zbMATH Keywords: conformal geometry
describes a project that uses: CLIFFORD
MaRDI profile type: Publication
full work available at URL: j.laa.2007.06.015
OpenAlex ID: W2086679389
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DOI: 10.1016/J.LAA.2007.06.015
description: scientific article; zbMATH DE number 5307056