NCMatrix % NCGroebnerBasis -- 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.
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

o10 = | z^2*x+z*y*z+z*x*y+y*z*y+y^2*x+y*x*z+x*z^2+x*y^2+x^3 z^2*y+z*y*x+z*x*z+y*z^2+y^3+y*x^2+x*z*x+x*y*z+x^2*y z^3+z*y^2+z*x^2+y*z*x+y^2*z+y*x*y+x*z*y+x*y*x+x^2*z |
      | z^2*y+z*y*x+z*x*z+y*z^2+y^3+y*x^2+x*z*x+x*y*z+x^2*y z^3+z*y^2+z*x^2+y*z*x+y^2*z+y*x*y+x*z*y+x*y*x+x^2*z z^2*x+z*y*z+z*x*y+y*z*y+y^2*x+y*x*z+x*z^2+x*y^2+x^3 |
      | z^3+z*y^2+z*x^2+y*z*x+y^2*z+y*x*y+x*z*y+x*y*x+x^2*z z^2*x+z*y*z+z*x*y+y*z*y+y^2*x+y*x*z+x*z^2+x*y^2+x^3 z^2*y+z*y*x+z*x*z+y*z^2+y^3+y*x^2+x*z*x+x*y*z+x^2*y |

o10 : NCMatrix

i11 : N3red = N3 % Igb

o11 = | -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3  2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3                      |
      | y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3  2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3                      -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y |
      | 2*y^2*z+y^3+y*x*y+x*y*x+2*x^2*z+x^3                      -y^2*z+y^3+y*x*z-y*x*y+x*y*z+x*y^2+2*x*y*x+x^2*z+3*x^2*y y^2*z+y*x*z+2*y*x*y+x*y*z+3*x*y^2-x*y*x-x^2*z+x^2*y+x^3  |

o11 : NCMatrix