sage: (x*y).inner_product(z) == y.inner_product(x*z)
True
- Our basis is normalized with respect to the natural inner product::
+ Our natural basis is normalized with respect to the natural inner
+ product unless we specify otherwise::
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
sage: all( b.norm() == 1 for b in J.gens() )
True
- Left-multiplication operators are symmetric because they satisfy
- the Jordan axiom::
+ Since our natural basis is normalized with respect to the natural
+ inner product, and since we know that this algebra is an EJA, any
+ left-multiplication operator's matrix will be symmetric because
+ natural->EJA basis representation is an isometry and within the EJA
+ the operator is self-adjoint by the Jordan axiom::
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
p = z**2 - 2
if p.is_irreducible():
field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
- S = [ s.change_ring(field) for s in S ]
+ S = [ s.change_ring(field) for s in S ]
self._basis_normalizers = tuple(
~(self.__class__.natural_inner_product(s,s).sqrt())
for s in S )
sage: (x*y).inner_product(z) == y.inner_product(x*z)
True
- Our basis is normalized with respect to the natural inner product::
+ Our natural basis is normalized with respect to the natural inner
+ product unless we specify otherwise::
sage: set_random_seed()
sage: n = ZZ.random_element(1,4)
sage: all( b.norm() == 1 for b in J.gens() )
True
- Left-multiplication operators are symmetric because they satisfy
- the Jordan axiom::
+ Since our natural basis is normalized with respect to the natural
+ inner product, and since we know that this algebra is an EJA, any
+ left-multiplication operator's matrix will be symmetric because
+ natural->EJA basis representation is an isometry and within the EJA
+ the operator is self-adjoint by the Jordan axiom::
sage: set_random_seed()
sage: n = ZZ.random_element(1,5)
p = z**2 - 2
if p.is_irreducible():
field = NumberField(p, 'sqrt2', embedding=RLF(2).sqrt())
- S = [ s.change_ring(field) for s in S ]
+ S = [ s.change_ring(field) for s in S ]
self._basis_normalizers = tuple(
~(self.__class__.natural_inner_product(s,s).sqrt())
for s in S )