class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
+ def conjugate_transpose(self):
+ r"""
+ Return the conjugate-transpose of this matrix.
+
+ SETUP::
+
+ sage: from mjo.hurwitz import HurwitzMatrixAlgebra
+
+ EXAMPLES::
+
+ sage: A = HurwitzMatrixAlgebra(2, QQbar, ZZ)
+ sage: M = A([ [ I, 2*I],
+ ....: [ 3*I, 4*I] ])
+ +------+------+
+ | -1*I | -3*I |
+ +------+------+
+ | -2*I | -4*I |
+ +------+------+
+
+ """
+ entries = [ [ self[j,i].conjugate()
+ for j in range(self.ncols())]
+ for i in range(self.nrows()) ]
+ return self.parent()._element_constructor_(entries)
+
def is_hermitian(self):
r"""
EXAMPLES::
- sage: A = HurwitzMatrixAlgebra(QQbar, ZZ, 2)
+ sage: A = HurwitzMatrixAlgebra(2, QQbar, ZZ)
sage: M = A([ [ 0,I],
....: [-I,0] ])
sage: M.is_hermitian()
True
"""
+ # A tiny bit faster than checking equality with the conjugate
+ # transpose.
return all( self[i,j] == self[j,i].conjugate()
for i in range(self.nrows())
for j in range(self.ncols()) )
"""
Element = HurwitzMatrixAlgebraElement
- def __init__(self, entry_algebra, scalars, n, **kwargs):
+ def __init__(self, n, entry_algebra, scalars, **kwargs):
from sage.rings.all import RR
if not scalars.is_subring(RR):
# Not perfect, but it's what we're using.
raise ValueError("scalar field is not real")
- super().__init__(entry_algebra, scalars, n, **kwargs)
+ super().__init__(n, entry_algebra, scalars, **kwargs)
def entry_algebra_gens(self):
r"""
sets of generators have cartinality 1,2,4, and 8 as you'd
expect::
- sage: HurwitzMatrixAlgebra(AA, AA, 2).entry_algebra_gens()
+ sage: HurwitzMatrixAlgebra(2, AA, AA).entry_algebra_gens()
(1,)
- sage: HurwitzMatrixAlgebra(QQbar, AA, 2).entry_algebra_gens()
+ sage: HurwitzMatrixAlgebra(2, QQbar, AA).entry_algebra_gens()
(1, I)
sage: Q = QuaternionAlgebra(AA,-1,-1)
- sage: HurwitzMatrixAlgebra(Q, AA, 2).entry_algebra_gens()
+ sage: HurwitzMatrixAlgebra(2, Q, AA).entry_algebra_gens()
(1, i, j, k)
sage: O = Octonions()
- sage: HurwitzMatrixAlgebra(O, AA, 2).entry_algebra_gens()
+ sage: HurwitzMatrixAlgebra(2, O, AA).entry_algebra_gens()
(e0, e1, e2, e3, e4, e5, e6, e7)
"""
SETUP::
- sage: from mjo.hurwitz import OctonionMatrixAlgebra
+ sage: from mjo.hurwitz import Octonions, OctonionMatrixAlgebra
EXAMPLES::
sage: OctonionMatrixAlgebra(3)
Module of 3 by 3 matrices with entries in Octonion algebra with base
ring Algebraic Real Field over the scalar ring Algebraic Real Field
- sage: OctonionMatrixAlgebra(3,QQ)
- Module of 3 by 3 matrices with entries in Octonion algebra with base
- ring Rational Field over the scalar ring Rational Field
+
+ ::
+
+ sage: OctonionMatrixAlgebra(3,scalars=QQ)
+ Module of 3 by 3 matrices with entries in Octonion algebra with
+ base ring Rational Field over the scalar ring Rational Field
+
+ ::
+
+ sage: O = Octonions(RR)
+ sage: A = OctonionMatrixAlgebra(1,O)
+ sage: A
+ Module of 1 by 1 matrices with entries in Octonion algebra with
+ base ring Real Field with 53 bits of precision over the scalar
+ ring Algebraic Real Field
+ sage: A.one()
+ +---------------------+
+ | 1.00000000000000*e0 |
+ +---------------------+
+ sage: A.gens()
+ (+---------------------+
+ | 1.00000000000000*e0 |
+ +---------------------+,
+ +---------------------+
+ | 1.00000000000000*e1 |
+ +---------------------+,
+ +---------------------+
+ | 1.00000000000000*e2 |
+ +---------------------+,
+ +---------------------+
+ | 1.00000000000000*e3 |
+ +---------------------+,
+ +---------------------+
+ | 1.00000000000000*e4 |
+ +---------------------+,
+ +---------------------+
+ | 1.00000000000000*e5 |
+ +---------------------+,
+ +---------------------+
+ | 1.00000000000000*e6 |
+ +---------------------+,
+ +---------------------+
+ | 1.00000000000000*e7 |
+ +---------------------+)
::
::
- sage: A1 = OctonionMatrixAlgebra(1,QQ)
- sage: A2 = OctonionMatrixAlgebra(1,QQ)
+ sage: A1 = OctonionMatrixAlgebra(1,scalars=QQ)
+ sage: A2 = OctonionMatrixAlgebra(1,scalars=QQ)
sage: cartesian_product([A1,A2])
Module of 1 by 1 matrices with entries in Octonion algebra with
base ring Rational Field over the scalar ring Rational Field (+)
True
"""
- def __init__(self, n, scalars=AA, prefix="E", **kwargs):
- super().__init__(Octonions(field=scalars),
+ def __init__(self, n, entry_algebra=None, scalars=AA, **kwargs):
+ if entry_algebra is None:
+ entry_algebra = Octonions(field=scalars)
+ super().__init__(n,
+ entry_algebra,
scalars,
- n,
- prefix=prefix,
**kwargs)
class QuaternionMatrixAlgebra(HurwitzMatrixAlgebra):
Module of 3 by 3 matrices with entries in Quaternion
Algebra (-1, -1) with base ring Algebraic Real Field
over the scalar ring Algebraic Real Field
- sage: QuaternionMatrixAlgebra(3,QQ)
+
+ ::
+
+ sage: QuaternionMatrixAlgebra(3,scalars=QQ)
Module of 3 by 3 matrices with entries in Quaternion
Algebra (-1, -1) with base ring Rational Field over
the scalar ring Rational Field
+ ::
+
+ sage: Q = QuaternionAlgebra(RDF, -1, -1)
+ sage: A = QuaternionMatrixAlgebra(1,Q)
+ sage: A
+ Module of 1 by 1 matrices with entries in Quaternion Algebra
+ (-1.0, -1.0) with base ring Real Double Field over the scalar
+ ring Algebraic Real Field
+ sage: A.one()
+ +-----+
+ | 1.0 |
+ +-----+
+ sage: A.gens()
+ (+-----+
+ | 1.0 |
+ +-----+,
+ +---+
+ | i |
+ +---+,
+ +---+
+ | j |
+ +---+,
+ +---+
+ | k |
+ +---+)
+
::
sage: A = QuaternionMatrixAlgebra(2)
::
- sage: A1 = QuaternionMatrixAlgebra(1,QQ)
- sage: A2 = QuaternionMatrixAlgebra(2,QQ)
+ sage: A1 = QuaternionMatrixAlgebra(1,scalars=QQ)
+ sage: A2 = QuaternionMatrixAlgebra(2,scalars=QQ)
sage: cartesian_product([A1,A2])
Module of 1 by 1 matrices with entries in Quaternion Algebra
(-1, -1) with base ring Rational Field over the scalar ring
True
"""
- def __init__(self, n, scalars=AA, **kwargs):
- # The -1,-1 gives us the "usual" definition of quaternion
- from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra
- Q = QuaternionAlgebra(scalars,-1,-1)
- super().__init__(Q, scalars, n, **kwargs)
+ def __init__(self, n, entry_algebra=None, scalars=AA, **kwargs):
+ if entry_algebra is None:
+ # The -1,-1 gives us the "usual" definition of quaternion
+ from sage.algebras.quatalg.quaternion_algebra import (
+ QuaternionAlgebra
+ )
+ entry_algebra = QuaternionAlgebra(scalars,-1,-1)
+ super().__init__(n, entry_algebra, scalars, **kwargs)
class ComplexMatrixAlgebra(HurwitzMatrixAlgebra):
sage: ComplexMatrixAlgebra(3)
Module of 3 by 3 matrices with entries in Algebraic Field
over the scalar ring Algebraic Real Field
- sage: ComplexMatrixAlgebra(3,QQ)
+
+ ::
+
+ sage: ComplexMatrixAlgebra(3,scalars=QQ)
Module of 3 by 3 matrices with entries in Algebraic Field
over the scalar ring Rational Field
+ ::
+
+ sage: A = ComplexMatrixAlgebra(1,CC)
+ sage: A
+ Module of 1 by 1 matrices with entries in Complex Field with
+ 53 bits of precision over the scalar ring Algebraic Real Field
+ sage: A.one()
+ +------------------+
+ | 1.00000000000000 |
+ +------------------+
+ sage: A.gens()
+ (+------------------+
+ | 1.00000000000000 |
+ +------------------+,
+ +--------------------+
+ | 1.00000000000000*I |
+ +--------------------+)
+
::
sage: A = ComplexMatrixAlgebra(2)
::
- sage: A1 = ComplexMatrixAlgebra(1,QQ)
- sage: A2 = ComplexMatrixAlgebra(2,QQ)
+ sage: A1 = ComplexMatrixAlgebra(1,scalars=QQ)
+ sage: A2 = ComplexMatrixAlgebra(2,scalars=QQ)
sage: cartesian_product([A1,A2])
Module of 1 by 1 matrices with entries in Algebraic Field over
the scalar ring Rational Field (+) Module of 2 by 2 matrices with
True
"""
- def __init__(self, n, scalars=AA, **kwargs):
- from sage.rings.all import QQbar
- super().__init__(QQbar, scalars, n, **kwargs)
+ def __init__(self, n, entry_algebra=None, scalars=AA, **kwargs):
+ if entry_algebra is None:
+ from sage.rings.all import QQbar
+ entry_algebra = QQbar
+ super().__init__(n, entry_algebra, scalars, **kwargs)
projects / sage.d.git / blobdiff