This command tests equality for matrices. If one of the inputs is an integer, then the test only will work if the integer is zero. Below, we test the well-definedness of the exponentiation operation using Groebner bases.
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/M2-12186-0/0.init -on-error exit --silent > /var/folders/46/9b86vqxj4hjcngvy7kd7sb140000gn/T/M2-12186-0/3.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 : Nred = N^3 % Igb | 2 3 2 2 2 2 2 2 2 3 2 3 2 3 | o10 = | -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 | o10 : NCMatrix |
i11 : B = A/I o11 = B o11 : NCQuotientRing |
i12 : phi = ncMap(B,A,gens B) o12 = NCRingMap B <--- A o12 : NCRingMap |
i13 : NB = phi N o13 = | x y z | | | | y z x | | | | z x y | o13 : NCMatrix |
i14 : N3B = NB^3 | 2 3 2 2 2 2 2 2 2 3 2 3 2 3 | o14 = | -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 | o14 : NCMatrix |
i15 : (phi Nred) == N3B o15 = true |