- Test the first polarization identity from my notes, Koecher Chapter
- III, or from Baes (2.3)::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: Lx = x.operator_matrix()
- sage: Ly = y.operator_matrix()
- sage: Lxx = (x*x).operator_matrix()
- sage: Lxy = (x*y).operator_matrix()
- sage: bool(2*Lx*Lxy + Ly*Lxx == 2*Lxy*Lx + Lxx*Ly)
- True
-
- Test the second polarization identity from my notes or from
- Baes (2.4)::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: x = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
- sage: Lx = x.operator_matrix()
- sage: Ly = y.operator_matrix()
- sage: Lz = z.operator_matrix()
- sage: Lzy = (z*y).operator_matrix()
- sage: Lxy = (x*y).operator_matrix()
- sage: Lxz = (x*z).operator_matrix()
- sage: bool(Lx*Lzy + Lz*Lxy + Ly*Lxz == Lzy*Lx + Lxy*Lz + Lxz*Ly)
- True
-
- Test the third polarization identity from my notes or from
- Baes (2.5)::
-
- sage: set_random_seed()
- sage: J = random_eja()
- sage: u = J.random_element()
- sage: y = J.random_element()
- sage: z = J.random_element()
- sage: Lu = u.operator_matrix()
- sage: Ly = y.operator_matrix()
- sage: Lz = z.operator_matrix()
- sage: Lzy = (z*y).operator_matrix()
- sage: Luy = (u*y).operator_matrix()
- sage: Luz = (u*z).operator_matrix()
- sage: Luyz = (u*(y*z)).operator_matrix()
- sage: lhs = Lu*Lzy + Lz*Luy + Ly*Luz
- sage: rhs = Luyz + Ly*Lu*Lz + Lz*Lu*Ly
- sage: bool(lhs == rhs)
- True
-