class HurwitzMatrixAlgebraElement(MatrixAlgebraElement):
+ def conjugate_transpose(self):
+ r"""
+ Return the conjugate-transpose of this matrix.
+
+ SETUP::
+
+ sage: from mjo.hurwitz import ComplexMatrixAlgebra
+
+ EXAMPLES::
+
+ sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
+ sage: M = A([ [ I, 2*I],
+ ....: [ 3*I, 4*I] ])
+ sage: M.conjugate_transpose()
+ +------+------+
+ | -I | -3*I |
+ +------+------+
+ | -2*I | -4*I |
+ +------+------+
+ sage: M.conjugate_transpose().to_vector()
+ (0, -1, 0, -3, 0, -2, 0, -4)
+
+ """
+ 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"""
SETUP::
- sage: from mjo.hurwitz import HurwitzMatrixAlgebra
+ sage: from mjo.hurwitz import (ComplexMatrixAlgebra,
+ ....: HurwitzMatrixAlgebra)
EXAMPLES::
- sage: A = HurwitzMatrixAlgebra(2, QQbar, ZZ)
+ sage: A = ComplexMatrixAlgebra(2, QQbar, ZZ)
sage: M = A([ [ 0,I],
....: [-I,0] ])
sage: M.is_hermitian()
True
+ ::
+
+ sage: A = HurwitzMatrixAlgebra(2, AA, QQ)
+ sage: M = A([ [1, 1],
+ ....: [1, 1] ])
+ 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()) )
+---------------------+
| 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 |
+ +---------------------+)
::
+-----+
| 1.0 |
+-----+
+ sage: A.gens()
+ (+-----+
+ | 1.0 |
+ +-----+,
+ +---+
+ | i |
+ +---+,
+ +---+
+ | j |
+ +---+,
+ +---+
+ | k |
+ +---+)
::
entry_algebra = QuaternionAlgebra(scalars,-1,-1)
super().__init__(n, entry_algebra, scalars, **kwargs)
+ def _entry_algebra_element_to_vector(self, entry):
+ r"""
+
+ SETUP::
+
+ sage: from mjo.hurwitz import QuaternionMatrixAlgebra
+
+ EXAMPLES::
+
+ sage: A = QuaternionMatrixAlgebra(2)
+ sage: u = A.entry_algebra().one()
+ sage: A._entry_algebra_element_to_vector(u)
+ (1, 0, 0, 0)
+ sage: i,j,k = A.entry_algebra().gens()
+ sage: A._entry_algebra_element_to_vector(i)
+ (0, 1, 0, 0)
+ sage: A._entry_algebra_element_to_vector(j)
+ (0, 0, 1, 0)
+ sage: A._entry_algebra_element_to_vector(k)
+ (0, 0, 0, 1)
+
+ """
+ from sage.modules.free_module import FreeModule
+ d = len(self.entry_algebra_gens())
+ V = FreeModule(self.entry_algebra().base_ring(), d)
+ return V(entry.coefficient_tuple())
class ComplexMatrixAlgebra(HurwitzMatrixAlgebra):
r"""
+------------------+
| 1.00000000000000 |
+------------------+
+ sage: A.gens()
+ (+------------------+
+ | 1.00000000000000 |
+ +------------------+,
+ +--------------------+
+ | 1.00000000000000*I |
+ +--------------------+)
::
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 |
+ +---------+------+
::
from sage.rings.all import QQbar
entry_algebra = QQbar
super().__init__(n, entry_algebra, scalars, **kwargs)
+
+ def _entry_algebra_element_to_vector(self, entry):
+ r"""
+
+ SETUP::
+
+ sage: from mjo.hurwitz import ComplexMatrixAlgebra
+
+ EXAMPLES::
+
+ sage: A = ComplexMatrixAlgebra(2, QQbar, QQ)
+ sage: A._entry_algebra_element_to_vector(QQbar(1))
+ (1, 0)
+ sage: A._entry_algebra_element_to_vector(QQbar(I))
+ (0, 1)
+
+ """
+ from sage.modules.free_module import FreeModule
+ d = len(self.entry_algebra_gens())
+ V = FreeModule(self.entry_algebra().base_ring(), d)
+ return V((entry.real(), entry.imag()))