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NCAlgebra :: NCMatrix % NCGroebnerBasis

NCMatrix % NCGroebnerBasis -- Reduces the entries of an NCMatrix with respect to an NCGroebnerBasis

Synopsis

Description

This command reduces the entries of an NCMatrix with respect to an NCGroebnerBasis.

i1 : A = QQ{x,y,z}

o1 = A

o1 : NCPolynomialRing
i2 : f = y*z + z*y - x^2

            2
o2 = zy+yz-x

o2 : A
i3 : g = x*z + z*x - y^2

         2
o3 = zx-y +xz

o3 : A
i4 : h = z^2 - x*y - y*x

      2
o4 = z -yx-xy

o4 : A
i5 : I = ncIdeal {f,g,h}

                             2      2      2
o5 = Two-sided ideal {zy+yz-x , zx-y +xz, z -yx-xy}

o5 : NCIdeal
i6 : Igb = ncGroebnerBasis I
--Calling Bergman for NCGB calculation.
--running: bergman -i /var/folders/46/9b86vqxj4hjcngvy7kd7sb140000gn/T/M2-12323-0/0.init -on-error exit --silent > /var/folders/46/9b86vqxj4hjcngvy7kd7sb140000gn/T/M2-12323-0/3.ter ... Complete!

      2    2                2
o6 = y x-xy ; Lead Term = (y x, 1)
       2  2                  2
     yx -x y; Lead Term = (yx , 1)
         2
     zx-y +xz; Lead Term = (zx, 1)
            2
     zy+yz-x ; Lead Term = (zy, 1)
      2                      2
     z -yx-xy; Lead Term = (z , 1)

o6 : NCGroebnerBasis
i7 : M = ncMatrix {{x, y, z}}

o7 = | x  y  z |

o7 : NCMatrix
i8 : sigma = ncMap(A,A,{y,z,x})

o8 = NCRingMap A <--- A

o8 : NCRingMap
i9 : N = ncMatrix {{M},{sigma M}, {sigma sigma M}}

o9 = | x  y  z |
     |         |
     | y  z  x |
     |         |
     | z  x  y |

o9 : NCMatrix
i10 : N3 = N^3

      |  2               2        2   2  3   2            2  3   2          2    3   2   2      2               2  |
o10 = | z x+zyz+zxy+yzy+y x+yxz+xz +xy +x   z y+zyx+zxz+yz +y +yx +xzx+xyz+x y  z +zy +zx +yzx+y z+yxy+xzy+xyx+x z |
      |                                                                                                            |
      |  2            2  3   2          2    3   2   2      2               2    2               2        2   2  3 |
      | z y+zyx+zxz+yz +y +yx +xzx+xyz+x y  z +zy +zx +yzx+y z+yxy+xzy+xyx+x z  z x+zyz+zxy+yzy+y x+yxz+xz +xy +x  |
      |                                                                                                            |
      |  3   2   2      2               2    2               2        2   2  3   2            2  3   2          2  |
      | z +zy +zx +yzx+y z+yxy+xzy+xyx+x z  z x+zyz+zxy+yzy+y x+yxz+xz +xy +x   z y+zyx+zxz+yz +y +yx +xzx+xyz+x y |

o10 : NCMatrix
i11 : N3red = N3 % Igb

      |   2   3               2       2    2     2                  2      2   2   3           2   3           2   3        |
o11 = | -y z+y +yxz-yxy+xyz+xy +2xyx+x z+3x y   y z+yxz+2yxy+xyz+3xy -xyx-x z+x y+x          2y z+y +yxy+xyx+2x z+x         |
      |                                                                                                                     |
      |   2                  2      2   2   3           2   3           2   3           2   3               2       2    2  |
      |  y z+yxz+2yxy+xyz+3xy -xyx-x z+x y+x          2y z+y +yxy+xyx+2x z+x          -y z+y +yxz-yxy+xyz+xy +2xyx+x z+3x y |
      |                                                                                                                     |
      |          2   3           2   3           2   3               2       2    2     2                  2      2   2   3 |
      |        2y z+y +yxy+xyx+2x z+x          -y z+y +yxz-yxy+xyz+xy +2xyx+x z+3x y   y z+yxz+2yxy+xyz+3xy -xyx-x z+x y+x  |

o11 : NCMatrix