Minkowski reduction of integral matrices
Author:
John L. Donaldson
Journal:
Math. Comp. 33 (1979), 201216
MSC:
Primary 10E25; Secondary 15A36, 68C05
MathSciNet review:
514819
Fulltext PDF Free Access
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Abstract: In 1905 Hermann Minkowski introduced his theory of reduction of positive definite quadratic forms. Recently, Hans J. Zassenhaus has suggested that this theory can be applied to the problem of row reduction of matrices of integers. Computational investigations have shown that for matrices with more columns than rows, the number of steps required for reduction decreases drastically. In this paper it is proved that as the number of columns increases, the probability that a matrix is Minkowski reduced approaches one. This fact is the motivation behind the introduction of a modified version of Minkowski reduction, resulting in a reduction procedure more suitable for computation.
 [1]
HERMANN MINKOWSKI, Gesammelte Abhandlungen. II, pp. 53100.
 [2]
HANS ZASSENHAUS, "Bilinear spaces and reduction," Unpublished manuscript.
 [3]
C. HERMITE, J. Reine Angew. Math., v. 41, 1851, pp. 191216.
 [4]
C. C. MacDUFFEE, The Theory of Matrices, Chelsea, New York, 1956.
 [5]
Gordon
H. Bradley, Algorithms for Hermite and Smith
normal matrices and linear Diophantine equations, Math. Comp. 25 (1971), 897–907. MR 0301909
(46 #1064), http://dx.doi.org/10.1090/S0025571819710301909X
 [6]
J.
Barkley Rosser, A method of computing exact inverses of matrices
with integer coefficients, J. Research Nat. Bur. Standards
49 (1952), 349–358. MR 0055796
(14,1128a)
 [7]
W. A. BLANKINSHIP, Comm. ACM, v. 9, 1966, p. 513.
 [8]
A.
Chatelet, Sur certains ensembles de tableaux et leur application
à la théorie des nombres, Ann. Sci. École Norm.
Sup. (3) 28 (1911), 105–202 (French). MR
1509137
 [9]
Hermann
Weyl, Theory of reduction for arithmetical
equivalence, Trans. Amer. Math. Soc. 48 (1940), 126–164.
MR
0002345 (2,35h), http://dx.doi.org/10.1090/S00029947194000023452
 [10]
G.
J. O. Jameson, Topology and normed spaces, Chapman and Hall,
London; Halsted Press [John Wiley & Sons], New York, 1974. MR 0463890
(57 #3828)
 [11]
P.
W. Aitchison, Two finiteness theorems in the Minkowski theory of
reduction, J. Austral. Math. Soc. 14 (1972),
336–351. MR 0318066
(47 #6615)
 [12]
S. S. RYSKOV, Soviet Math. Dokl., v. 12, 1971, pp. 946950.
 [13]
A.
Rényi, Probability theory, NorthHolland Publishing
Co., AmsterdamLondon; American Elsevier Publishing Co., Inc., New York,
1970. Translated by László Vekerdi; NorthHolland Series in
Applied Mathematics and Mechanics, Vol. 10. MR 0315747
(47 #4296)
 [14]
B.
L. van der Waerden, Die Reduktionstheorie der positiven
quadratischen Formen, Acta Math. 96 (1956),
265–309 (German). MR 0082513
(18,562e)
 [15]
Claude
Chevalley, Fundamental concepts of algebra, Academic Press
Inc., New York, 1956. MR 0082459
(18,553a)
 [16]
P. TAMMELA, Soviet Math. Dokl., v. 14, 1973, p. 651.
 [17]
Marvin
Marcus, Finite dimensional multilinear algebra. Part 1, Marcel
Dekker, Inc., New York, 1973. Pure and Applied Mathematics, Vol. 23. MR 0352112
(50 #4599)
 [1]
 HERMANN MINKOWSKI, Gesammelte Abhandlungen. II, pp. 53100.
 [2]
 HANS ZASSENHAUS, "Bilinear spaces and reduction," Unpublished manuscript.
 [3]
 C. HERMITE, J. Reine Angew. Math., v. 41, 1851, pp. 191216.
 [4]
 C. C. MacDUFFEE, The Theory of Matrices, Chelsea, New York, 1956.
 [5]
 G. H. BRADLEY, Math. Comp., v. 25, 1971, pp. 897907. MR 0301909 (46:1064)
 [6]
 J. B. ROSSER, J. Res. Nat. Bur. Standards, v. 49, 1952, pp. 349358. MR 0055796 (14:1128a)
 [7]
 W. A. BLANKINSHIP, Comm. ACM, v. 9, 1966, p. 513.
 [8]
 A. CHATELET, Ann. Ecole Norm. Ill, v. 28, 1911, pp. 105202. MR 1509137
 [9]
 H. WEYL, Trans. Amer. Math. Soc., v. 48, 1940, pp. 126164. MR 0002345 (2:35h)
 [10]
 G. J. O. JAMESON, Topology and Normed Spaces, Chapman and Hall, London, 1974. MR 0463890 (57:3828)
 [11]
 P. W. AITCHISON, J. Austral. Math. Soc., v. 14, 1972, pp. 336351. MR 0318066 (47:6615)
 [12]
 S. S. RYSKOV, Soviet Math. Dokl., v. 12, 1971, pp. 946950.
 [13]
 A. RENYI, Probability Theory, NorthHolland, Amsterdam and London, 1970. MR 0315747 (47:4296)
 [14]
 B. van DER WAERDEN, Acta Math., v. 96, 1956, pp. 265309. MR 0082513 (18:562e)
 [15]
 C. CHEVALLEY, Fundamental Concepts of Algebra, Academic Press, New York, 1956. MR 0082459 (18:553a)
 [16]
 P. TAMMELA, Soviet Math. Dokl., v. 14, 1973, p. 651.
 [17]
 MARVIN MARCUS, Finite Dimensional Multilinear Algebra, Part 1, Dekker, New York, 1973. MR 0352112 (50:4599)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197905148197
PII:
S 00255718(1979)05148197
Article copyright:
© Copyright 1979
American Mathematical Society
