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SNF - Smith Normalform - Elementarteiler ℤ

SNF 4x4 ℤ

SNF 4x4  ℤ
Smtih-Normalform with Elementar-Matrices to build row-operation P (extended on left side) and column-operations Q (extended on right side)
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] )