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Journal Article

Algorithms for Hermite and Smith Normal Matrices and Linear Diophantine Equations

Gordon H. Bradley
Mathematics of Computation
Vol. 25, No. 116 (Oct., 1971), pp. 897-907
DOI: 10.2307/2004354
Stable URL: http://www.jstor.org/stable/2004354
Page Count: 11
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Algorithms for Hermite and Smith Normal Matrices and Linear Diophantine Equations
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Abstract

New algorithms for constructing the Hermite normal form (triangular) and Smith normal form (diagonal) of an integer matrix are presented. A new algorithm for determining the set of solutions to a system of linear diophantine equations is presented. A modification of the Hermite algorithm gives an integer-preserving algorithm for solving linear equations with real-valued variables. Rough bounds for the number of operations are cubic polynomials involving the order of the matrix and the determinant of the matrix. The algorithms are valid if the elements of the matrix are in a principal ideal domain.

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