Bilinear Forms Derived from Lipschitzian Elements in Clifford Algebras

Bilinear Forms Derived from Lipschitzian Elements in Clifford Algebras In every Clifford algebra $${\mathrm {Cl}}(V,q)$$ Cl ( V , q ) , there is a Lipschitz monoid (or semi-group) $${\mathrm {Lip}}(V,q)$$ Lip ( V , q ) , which is in most cases the monoid generated by the vectors of V. This monoid is useful for many reasons, not only because of the natural homomorphism from the group $${\mathrm {GLip}}(V,q)$$ GLip ( V , q ) of its invertible elements onto the group $${\mathrm {O}}(V,q)$$ O ( V , q ) of orthogonal transformations. From every non-zero $$a\in {\mathrm {Lip}}(V,q)$$ a ∈ Lip ( V , q ) , we can derive a bilinear form $$\phi$$ ϕ on the support S of a in V; it is q-compatible: $$\phi (x,x)=q(x)$$ ϕ ( x , x ) = q ( x ) for all $$x\in S$$ x ∈ S . Conversely, every q-compatible bilinear form on a subspace S of V can be derived from an element $$a\in {\mathrm {Lip}}(V,q)$$ a ∈ Lip ( V , q ) which is unique up to an invertible scalar; and a is invertible if and only if $$\phi$$ ϕ is non-degenerate. This article studies the relations between a, $$\phi$$ ϕ and (when a is invertible) the orthogonal transformation g derived from a; it provides both theoretical knowledge and algorithms. It provides an effective tool for the factorization of lipschitzian elements, based on this theorem: if $$(v_1,v_2,\ldots ,v_s)$$ ( v 1 , v 2 , … , v s ) is a basis of S, then $$a=\kappa \,v_1v_2\ldots v_s$$ a = κ v 1 v 2 … v s (for some invertible scalar $$\kappa$$ κ ) if and only if the matrix of $$\phi$$ ϕ in this basis is lower triangular. This theorem is supported by an algorithm of triangularization of bilinear forms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Applied Clifford Algebras Springer Journals

Bilinear Forms Derived from Lipschitzian Elements in Clifford Algebras

, Volume 28 (1) – Feb 24, 2018
37 pages

/lp/springer_journal/bilinear-forms-derived-from-lipschitzian-elements-in-clifford-algebras-yuebut1wf6
Publisher
Springer International Publishing
Copyright © 2018 by Springer International Publishing AG, part of Springer Nature
Subject
Physics; Mathematical Methods in Physics; Theoretical, Mathematical and Computational Physics; Applications of Mathematics; Physics, general
ISSN
0188-7009
eISSN
1661-4909
D.O.I.
10.1007/s00006-018-0842-2
Publisher site
See Article on Publisher Site

Abstract

In every Clifford algebra $${\mathrm {Cl}}(V,q)$$ Cl ( V , q ) , there is a Lipschitz monoid (or semi-group) $${\mathrm {Lip}}(V,q)$$ Lip ( V , q ) , which is in most cases the monoid generated by the vectors of V. This monoid is useful for many reasons, not only because of the natural homomorphism from the group $${\mathrm {GLip}}(V,q)$$ GLip ( V , q ) of its invertible elements onto the group $${\mathrm {O}}(V,q)$$ O ( V , q ) of orthogonal transformations. From every non-zero $$a\in {\mathrm {Lip}}(V,q)$$ a ∈ Lip ( V , q ) , we can derive a bilinear form $$\phi$$ ϕ on the support S of a in V; it is q-compatible: $$\phi (x,x)=q(x)$$ ϕ ( x , x ) = q ( x ) for all $$x\in S$$ x ∈ S . Conversely, every q-compatible bilinear form on a subspace S of V can be derived from an element $$a\in {\mathrm {Lip}}(V,q)$$ a ∈ Lip ( V , q ) which is unique up to an invertible scalar; and a is invertible if and only if $$\phi$$ ϕ is non-degenerate. This article studies the relations between a, $$\phi$$ ϕ and (when a is invertible) the orthogonal transformation g derived from a; it provides both theoretical knowledge and algorithms. It provides an effective tool for the factorization of lipschitzian elements, based on this theorem: if $$(v_1,v_2,\ldots ,v_s)$$ ( v 1 , v 2 , … , v s ) is a basis of S, then $$a=\kappa \,v_1v_2\ldots v_s$$ a = κ v 1 v 2 … v s (for some invertible scalar $$\kappa$$ κ ) if and only if the matrix of $$\phi$$ ϕ in this basis is lower triangular. This theorem is supported by an algorithm of triangularization of bilinear forms.

Journal

Advances in Applied Clifford AlgebrasSpringer Journals

Published: Feb 24, 2018

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