PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 97, Number 1, May 1986 A SHORT PROOF OF THE EXISTENCE OF VECTOR EUCLIDEAN ALGORITHMS HELAMAN FERGUSON ABSTRACT. The classical Euclidean algorithm for pairs of real numbers is generalized to real n-vectors by Alg(n, Z). An iteration of Alg(r¡, Z) is defined by three steps. Given n real numbers Alg(n, Z) 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 GL(n, Z) matrix or the simultaneous approximations will be rows of GL(n, Z) matrices constructed by Alg(rj, Z). This algorithm characterizes linear inde- pendence of reals over rationals by GL(n, Z) orbits of rank n — 1 matrices. Let x G R™ be a row vector, n > 1, Min, R) the n x n real matrices, and In the nxn identity matrix. An integral vector b G Z™ is a nearest integral vector to x G R" if the corresponding coordinate entries of b and x differ by no more than one half. Let A denote the transpose of the matrix A. Define the matrix norm of A by \A\ where |A|2 = Trace AA, and similarly define |i|. This norm is submultiplicative as well as subadditive. Define m G Z™ to be a relation for x if m ^ 0 and xrh = 0. The coordinates of x are said to be Z-linearly independent if x has no relation. If x ^ 0, set P = xxln — xx, then xP = 0 and rank P = n —1. Let GL(n, Z) be the integral nxn matrices with det = ±1. Any row or column of any GL(n, Z) matrix consists of relatively prime integers. GL(n, Z) acts on M(n,R) by multiplication on the left. The algorithm Alg(n, Z) will be defined by a single iteration which replaces a vector, matrix pair x, P by a pair xA~x,AP for the integral matrix A G GL(n, Z) as constructed in Steps 1„, 2„ and 3„ below. The following notation for x and P will be assumed in this inductive definition of Alg(n, Z), cf. Step 2n. Suppose x t¿ 0, xP = 0, rank P = n — 1 for a real nxn matrix P. If the last entry of x is ieR and t ^ 0, set x — (wr,t), u G Rn_1. Set P = [^] where v is the last row of P. Note that xP = 0 implies uW = —v. Alg(l, Z), n = 1. If x = 0, terminate; otherwise set A = 1 and replace x,P by x, P where P = 0. Alg(n, Z), n > 1. If some entry of x is zero, terminate; otherwise perform the following three steps. Step ln. Let the permutation matrix E exchange a smallest row of P with the last row of P. Replace x, P by xE^1, EP. Received by the editors March 5, 1985 and, in revised form, May 31, 1985. 1980 Mathematics Subject Classification. Primary 10E45, 10F10; Secondary 20H05, 10F20, 10M20. Key words and phrases. Euclidean algorithm, relations, simultaneous approximations, linear dependence, independence, GL(n,Z), orbits. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page 8 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use