Abstract
An integer Square matrix with a deteminant of + 1 or −1 is called unimodular. Given a (m,m) integer matrix A, there exist unimodular matrices U,K such that S(A)=UAK is a diagonal matrix with positive diagonal elements d1,...,dr (r:=rank(A)) and zero diagonal elements dr+1,...,dm. In particular di divides di+1 (i=1,...,r−1). This was proved by Smith [21] in 1861 and the matrix S(A) is known as the Smith normal form of A.
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Bachem, A., Kannan, R. (1979). Applications of Polynomial Smith Normal form Calculations. In: Collatz, L., Meinardus, G., Wetterling, W. (eds) Numerische Methoden bei graphentheoretischen und kombinatorischen Problemen. International Series of Numerical Mathematics / Internationale Schriftenreihe zur Numerischen Mathematik / Série Internationale D’Analyse Numérique, vol 46. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5997-4_1
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DOI: https://doi.org/10.1007/978-3-0348-5997-4_1
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