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Author: Juan Carlos Bustamante
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typeset_mode(True)

A = matrix(ZZ, [[1,0,12,16], [0,1,0,8], [1, 2, 6, 14]])
A.smith_form()
((100001000060)\displaystyle \left(\begin{array}{rrrr} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 6 & 0 \end{array}\right), (0010101−4−1)\displaystyle \left(\begin{array}{rrr} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & -4 & -1 \end{array}\right), (1−8−620010−8011−30001)\displaystyle \left(\begin{array}{rrrr} 1 & -8 & -6 & 20 \\ 0 & 1 & 0 & -8 \\ 0 & 1 & 1 & -3 \\ 0 & 0 & 0 & 1 \end{array}\right))

Exemple n.1, p 513

M= matrix(ZZ,[[6,8],[9,15]]) # ZZ pour dire que l'anneau des coefficients est celui des entiers.
M.smith_form()
((10018)\displaystyle \left(\begin{array}{rr} 1 & 0 \\ 0 & 18 \end{array}\right), (1−13−4)\displaystyle \left(\begin{array}{rr} 1 & -1 \\ 3 & -4 \end{array}\right), (2−7−13)\displaystyle \left(\begin{array}{rr} 2 & -7 \\ -1 & 3 \end{array}\right))

Le résultat est une liste de trois matrices.. Pour savoir ce qu'elles représentent, on peut faire M.smith_form?

Exemple 2.

Cette fois l'anneau des coefficients est Q[x]\mathbb{Q}[x], il faut le dire.

A = QQ['x']
M=matrix(A,[[x-1,0,1],[0,x-2,2],[0,0,x-3]])
M
(x−1010x−2200x−3)\displaystyle \left(\begin{array}{rrr} x - 1 & 0 & 1 \\ 0 & x - 2 & 2 \\ 0 & 0 & x - 3 \end{array}\right)
S = M.smith_form()[0]
S
(10001000x3−6x2+11x−6)\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & x^{3} - 6 x^{2} + 11 x - 6 \end{array}\right)
S[2,2].factor()
(x−3)⋅(x−2)⋅(x−1)\displaystyle (x - 3) \cdot (x - 2) \cdot (x - 1)

Exemple 3.

Cette fois avec les coefficients dans Z[i]\mathbb{Z}[i].

R = ZZ[I]
R # Pour afficher quelle est la définition de R.
I = R.basis()[1]
Order in Number Field in I with defining polynomial x^2 + 1 

M = matrix([[1+8*I,-5+I],[-23+2*I,13*I]])
M.parent() # Pour afficher le type d'objet que M est : l'anneau des coefficients.
M
Full MatrixSpace of 2 by 2 dense matrices over Order in Number Field in I with defining polynomial x^2 + 1
(8I+1I−52I−2313I)\displaystyle \left(\begin{array}{rr} 8 I + 1 & I - 5 \\ 2 I - 23 & 13 I \end{array}\right)
M.smith_form()
((1002I+200)\displaystyle \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 I + 2 \\ 0 & 0 \end{array}\right), (100I−1−44I+1−2I−4I+3−2I−5)\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ I - 1 & -4 & 4 I + 1 \\ -2 I & -4 I + 3 & -2 I - 5 \end{array}\right), (1−6I−801)\displaystyle \left(\begin{array}{rr} 1 & -6 I - 8 \\ 0 & 1 \end{array}\right))

Un autre exemple

Où en plus on fait les manipulations des matrices "pour vrai", c'est à dire avec les opérations élémentaires.

M=matrix([[1,8+6*I],[1+I,-4],[1-I,0]])
M
M.smith_form()
(16I+8I+1−4−I+10)\displaystyle \left(\begin{array}{rr} 1 & 6 I + 8 \\ I + 1 & -4 \\ -I + 1 & 0 \end{array}\right)
((1002I+200)\displaystyle \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 I + 2 \\ 0 & 0 \end{array}\right), (100I−1−44I+1−2I−4I+3−2I−5)\displaystyle \left(\begin{array}{rrr} 1 & 0 & 0 \\ I - 1 & -4 & 4 I + 1 \\ -2 I & -4 I + 3 & -2 I - 5 \end{array}\right), (1−6I−801)\displaystyle \left(\begin{array}{rr} 1 & -6 I - 8 \\ 0 & 1 \end{array}\right))
M= M.with_added_multiple_of_row(1,0,-(1+I))
M = M.with_added_multiple_of_row(2,0,-(1-I))
M
(16I+80−14I−602I−14)\displaystyle \left(\begin{array}{rr} 1 & 6 I + 8 \\ 0 & -14 I - 6 \\ 0 & 2 I - 14 \end{array}\right)
M=M.with_added_multiple_of_column(1,0,-(8+6*I))
M
(100−14I−602I−14)\displaystyle \left(\begin{array}{rr} 1 & 0 \\ 0 & -14 I - 6 \\ 0 & 2 I - 14 \end{array}\right)
M=M.with_rescaled_row(2,I)
M
(100−14I−60−14I−2)\displaystyle \left(\begin{array}{rr} 1 & 0 \\ 0 & -14 I - 6 \\ 0 & -14 I - 2 \end{array}\right)
M=M.with_added_multiple_of_row(2,1,-1)
M
(100−14I−604)\displaystyle \left(\begin{array}{rr} 1 & 0 \\ 0 & -14 I - 6 \\ 0 & 4 \end{array}\right)
M.with_added_multiple_of_row(1,2,-1-3*I)
(100−26I−1004)\displaystyle \left(\begin{array}{rr} 1 & 0 \\ 0 & -26 I - 10 \\ 0 & 4 \end{array}\right)


M=M.with_added_multiple_of_row(1,2,1+I)
M
(1002I−204)\displaystyle \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 I - 2 \\ 0 & 4 \end{array}\right)
M.with_added_multiple_of_row(2,1,(1+I))
(1002I−200)\displaystyle \left(\begin{array}{rr} 1 & 0 \\ 0 & 2 I - 2 \\ 0 & 0 \end{array}\right)