Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A short proof of the existence of vector Euclidean algorithms


Author: Helaman Ferguson
Journal: Proc. Amer. Math. Soc. 97 (1986), 8-10
MSC: Primary 11H46; Secondary 11J13, 11Y16
MathSciNet review: 831375
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The classical Euclidean algorithm for pairs of real numbers is generalized to real $ n$-vectors by $ {\text{Alg}}\left( {n,{\mathbf{Z}}} \right)$. An iteration of $ {\text{Alg}}\left( {n,{\mathbf{Z}}} \right)$ is defined by three steps. Given $ n$ real numbers $ {\text{Alg}}\left( {n,{\mathbf{Z}}} \right)$ constructs either $ n$ coefficients of a nontrivial integral linear combination which is zero or $ n$ independent sets of simultaneous approximations. Either the coefficients will be a column of a $ {\text{GL}}\left( {n,{\mathbf{Z}}} \right)$ matrix or the simultaneous approximations will be rows of $ {\text{GL}}\left( {n,{\mathbf{Z}}} \right)$ matrices constructed by $ {\text{Alg}}\left( {n,{\mathbf{Z}}} \right)$. This algorithm characterizes linear independence of reals over rationals by $ {\text{GL}}\left( {n,{\mathbf{Z}}} \right)$ orbits of rank $ n - 1$ matrices.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 11H46, 11J13, 11Y16

Retrieve articles in all journals with MSC: 11H46, 11J13, 11Y16


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1986-0831375-X
PII: S 0002-9939(1986)0831375-X
Keywords: Euclidean algorithm, relations, simultaneous approximations, linear dependence, independence, $ {\text{GL}}\left( {n,{\mathbf{Z}}} \right)$, orbits
Article copyright: © Copyright 1986 American Mathematical Society