SNF - Smith Normalform - Elementarteiler ℤ
SNF 4x4 ℤ
Start slider 0
Input row operation to (7) IP, add new elementar-matrix to left side
[p+] set index to next P
E(a,b,c,n) row: a = a + b*c, n row dimension
Input colum operation to (9) IQ, add new elementar-matrix to right side
[q+] set index to next Q
E(a,b,c,n) col: a*c + b = b, n col dimension
T(a,b,n) Exchange row/col a <-> b, n row/col dimension
Example
A:={ {9, -36, 30},{ -36, 192, -180}, {30, -180, 180}};
Li := {{0, 0}, {1, 0}, {2, 0}, {3, 0}, {3, 1}, {3, 2}, {4, 2}, {5, 2}, {6, 2} };
IP:{E(3,2,-3,3),E(2,3,-1,3),E(3,2,-3,3),T(1,3,3) E(1,3,-3,3), E(3,1,-3,3),E(2,1,4,3)};
IQ:{E(1,2,24,3),E(1,3,-30,3)};
Sage MathCell
https://sagecell.sagemath.org/
M=matrix(ZZ,[[-6, 111, -36, 6],[5, -672 ,210, 74],[0, -255, 81, 24],[-7,255,-81,-10]])
M.smith_form()
M=matrix(ZZ,[[9, -36, 30],[ -36, 192, -180], [30, -180, 180]])
M.smith_form()
M=matrix(ZZ,[[40, -20, 36, 12],[-24, -4 ,28, 48],[0, 4, 0, -8],[8,0,0,-8]])
M.smith_form()
(
[ 4 0 0 0] [ 1 0 0 0] [120 84 128 139]
[ 0 4 0 0] [ 0 1 0 0] [214 150 229 248]
[ 0 0 4 0] [ 0 0 1 0] [-50 -35 -53 -58]
[ 0 0 0 120] [-26 -18 -28 1] [107 75 114 124]
)