Two Different Computing Methods of the Smith Arithmetic Determinant

Authors: Xing-Jian Li, Shen Qu

Abstract:

The Smith arithmetic determinant is investigated in this paper. By using two different methods, we derive the explicit formula for the Smith arithmetic determinant.

Keywords: Elementary row transformation, Euler function, Matrix decomposition, Smith arithmetic determinant.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1089413

References:

[1] Tingzhu Huang, Junhua He and Yongbin Li, Advanced Algebra, Higer Education Press, Beijing, 2011.
[2] Chengdong Pan and Chengbiao Pan, Elementary Number Theory, Beijing University Press, Beijing, 1998.
[3] Brualdi and Richard A, Combinatorial Matrix Theory, Cambridge University Press, Cambridge, 1992.
[4] Beslin S and Ligh S, Greatest common divisor matrices(J), Linear Algebra Appl, 1989, 118:69-76.
[5] McCarthy P J, A generalized of Smith’s determinant(J), Canadian Math. Bull, 1986, 29:109-113.
[6] Beslin S and el Kassar N, GCD matrices and Smith’s determinant for a U.F.D.(J), Bull. Number Theory, 1989, 8:23-28.