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
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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.

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Keywords: Euclidean algorithm, relations, simultaneous approximations, linear dependence, independence, $ {\text{GL}}\left( {n,{\mathbf{Z}}} \right)$, orbits
Article copyright: © Copyright 1986 American Mathematical Society