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
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i2 : f = y*z + z*y - x^2
2
o2 = zy+yz-x
o2 : A
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i3 : g = x*z + z*x - y^2
2
o3 = zx-y +xz
o3 : A
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i4 : h = z^2 - x*y - y*x
2
o4 = z -yx-xy
o4 : A
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i5 : I = ncIdeal {f,g,h}
2 2 2
o5 = Two-sided ideal {zy+yz-x , zx-y +xz, z -yx-xy}
o5 : NCIdeal
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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
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i7 : M = ncMatrix {{x, y, z}}
o7 = | x y z |
o7 : NCMatrix
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i8 : sigma = ncMap(A,A,{y,z,x})
o8 = NCRingMap A <--- A
o8 : NCRingMap
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i9 : N = ncMatrix {{M},{sigma M}, {sigma sigma M}}
o9 = | x y z |
| y z x |
| z x y |
o9 : NCMatrix
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i10 : Nred = N^3 % Igb
o10 = | -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 |
o10 : NCMatrix
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i11 : B = A/I
o11 = B
o11 : NCQuotientRing
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i12 : phi = ncMap(B,A,gens B)
o12 = NCRingMap B <--- A
o12 : NCRingMap
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i13 : NB = phi N
o13 = | x y z |
| y z x |
| z x y |
o13 : NCMatrix
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i14 : N3B = NB^3
o14 = | -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 |
o14 : NCMatrix
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i15 : (phi Nred) == N3B
o15 = true
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