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
class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
+ def conjugate(self):
+ r"""
+ Return the entrywise conjugate of this matrix.
+
+ SETUP::
+
+ sage: from mjo.hurwitz import ComplexMatrixAlgebra
+
+ EXAMPLES::
+
+ sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
+ sage: M = A([ [ I, 1 + 2*I],
+ ....: [ 3*I, 4*I] ])
+ sage: M.conjugate()
+ +------+----------+
+ | -I | -2*I + 1 |
+ +------+----------+
+ | -3*I | -4*I |
+ +------+----------+
+
+ """
+ entries = [ [ self[i,j].conjugate()
+ for j in range(self.ncols())]
+ for i in range(self.nrows()) ]
+ return self.parent()._element_constructor_(entries)
+
def conjugate_transpose(self):
r"""
Return the conjugate-transpose of this matrix.
for j in range(self.ncols()) )
+ def is_skew_symmetric(self):
+ r"""
+ Return whether or not this matrix is skew-symmetric.
+
+ SETUP::
+
+ sage: from mjo.hurwitz import (ComplexMatrixAlgebra,
+ ....: HurwitzMatrixAlgebra)
+
+ EXAMPLES::
+
+ sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
+ sage: M = A([ [ 0,I],
+ ....: [-I,1] ])
+ sage: M.is_skew_symmetric()
+ False
+
+ ::
+
+ sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
+ sage: M = A([ [ 0, 1+I],
+ ....: [-1-I, 0] ])
+ sage: M.is_skew_symmetric()
+ True
+
+ ::
+
+ sage: A = HurwitzMatrixAlgebra(2, AA, QQ)
+ sage: M = A([ [1, 1],
+ ....: [1, 1] ])
+ sage: M.is_skew_symmetric()
+ False
+
+ ::
+
+ sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
+ sage: M = A([ [2*I , 1 + I],
+ ....: [-1 + I, -2*I] ])
+ sage: M.is_skew_symmetric()
+ False
+
+ """
+ # A tiny bit faster than checking equality with the negation
+ # of the transpose.
+ return all( self[i,j] == -self[j,i]
+ for i in range(self.nrows())
+ for j in range(self.ncols()) )
+
+
class HurwitzMatrixAlgebra(MatrixAlgebra):
r"""
A class of matrix algebras whose entries come from a Hurwitz
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
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