Conjugating twice gets you the original element::
- sage: set_random_seed()
sage: O = Octonions()
sage: x = O.random_element()
sage: x.conjugate().conjugate() == x
This method is idempotent::
- sage: set_random_seed()
sage: O = Octonions()
sage: x = O.random_element()
sage: x.real().real() == x.real()
This method is idempotent::
- sage: set_random_seed()
sage: O = Octonions()
sage: x = O.random_element()
sage: x.imag().imag() == x.imag()
The norm is nonnegative and belongs to the base field::
- sage: set_random_seed()
sage: O = Octonions()
sage: n = O.random_element().norm()
sage: n >= 0 and n in O.base_ring()
The norm is homogeneous::
- sage: set_random_seed()
sage: O = Octonions()
sage: x = O.random_element()
sage: alpha = O.base_ring().random_element()
TESTS::
- sage: set_random_seed()
sage: O = Octonions()
sage: x = O.random_element()
sage: x.is_zero() or ( x*x.inverse() == O.one() )
This gives the correct unit element::
- sage: set_random_seed()
sage: O = Octonions()
sage: x = O.random_element()
sage: x*O.one() == x and O.one()*x == x
TESTS::
- sage: set_random_seed()
sage: A = OctonionMatrixAlgebra(ZZ.random_element(10))
sage: x = A.random_element()
sage: x*A.one() == x and A.one()*x == x
TESTS::
- sage: set_random_seed()
sage: A = QuaternionMatrixAlgebra(ZZ.random_element(10))
sage: x = A.random_element()
sage: x*A.one() == x and A.one()*x == x
sage: (I,) = A.entry_algebra().gens()
sage: A([ [1+I, 1],
....: [-1, -I] ])
- +-------+----+
- | I + 1 | 1 |
- +-------+----+
- | -1 | -I |
- +-------+----+
+ +---------+------+
+ | 1 + 1*I | 1 |
+ +---------+------+
+ | -1 | -1*I |
+ +---------+------+
::
TESTS::
- sage: set_random_seed()
sage: A = ComplexMatrixAlgebra(ZZ.random_element(10))
sage: x = A.random_element()
sage: x*A.one() == x and A.one()*x == x