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group_element.py
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group_element.py
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"""
Matrix Group Elements
EXAMPLES::
sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,0],[0,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens); G
Matrix group over Finite Field of size 3 with 2 generators (
[1 0] [1 1]
[0 1], [0 1]
)
sage: g = G([[1,1],[0,1]])
sage: h = G([[1,2],[0,1]])
sage: g*h
[1 0]
[0 1]
You cannot add two matrices, since this is not a group operation.
You can coerce matrices back to the matrix space and add them
there::
sage: g + h
Traceback (most recent call last):
...
TypeError: unsupported operand type(s) for +:
'FinitelyGeneratedMatrixGroup_gap_with_category.element_class' and
'FinitelyGeneratedMatrixGroup_gap_with_category.element_class'
sage: g.matrix() + h.matrix()
[2 0]
[0 2]
Similarly, you cannot multiply group elements by scalars but you can
do it with the underlying matrices::
sage: 2*g
Traceback (most recent call last):
...
TypeError: unsupported operand parent(s) for '*': 'Integer Ring' and 'Matrix group over Finite Field of size 3 with 2 generators (
[1 0] [1 1]
[0 1], [0 1]
)'
AUTHORS:
- David Joyner (2006-05): initial version David Joyner
- David Joyner (2006-05): various modifications to address William
Stein's TODO's.
- William Stein (2006-12-09): many revisions.
- Volker Braun (2013-1) port to new Parent, libGAP.
"""
#*****************************************************************************
# Copyright (C) 2006 David Joyner and William Stein <wstein@gmail.com>
# Copyright (C) 2013 Volker Braun <vbraun.name@gmail.com>
#
# Distributed under the terms of the GNU General Public License (GPL)
#
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.interfaces.gap import gap
from sage.structure.element import MultiplicativeGroupElement
from sage.matrix.matrix import Matrix, is_Matrix
from sage.structure.factorization import Factorization
from sage.structure.element import have_same_parent
from sage.libs.gap.element import GapElement, GapElement_List
from sage.misc.cachefunc import cached_method
from sage.groups.libgap_wrapper import ElementLibGAP
from sage.groups.libgap_mixin import GroupElementMixinLibGAP
def is_MatrixGroupElement(x):
"""
Test whether ``x`` is a matrix group element
INPUT:
- ``x`` -- anything.
OUTPUT:
Boolean.
EXAMPLES::
sage: from sage.groups.matrix_gps.group_element import is_MatrixGroupElement
sage: is_MatrixGroupElement('helloooo')
False
sage: G = GL(2,3)
sage: is_MatrixGroupElement(G.an_element())
True
"""
return isinstance(x, MatrixGroupElement_base)
class MatrixGroupElement_base(MultiplicativeGroupElement):
"""
Base class for elements of matrix groups.
You should use one of the two subclasses:
* :class:`MatrixGroupElement_sage` implements the group
multiplication using Sage matrices.
* :class:`MatrixGroupElement_gap` implements the group
multiplication using libGAP matrices.
The base class only assumes that derived classes implement
:meth:`matrix`.
EXAMPLES::
sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,0],[0,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: g = G.random_element()
sage: type(g)
<class 'sage.groups.matrix_gps.group_element.FinitelyGeneratedMatrixGroup_gap_with_category.element_class'>
"""
def __hash__(self):
r"""
TESTS::
sage: MS = MatrixSpace(GF(3), 2)
sage: G = MatrixGroup([MS([1,1,0,1]), MS([1,0,1,1])])
sage: g = G.an_element()
sage: hash(g)
0
"""
return hash(self.matrix())
def _repr_(self):
"""
Return string representation of this matrix.
EXAMPLES::
sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,0],[0,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: g = G([[1, 1], [0, 1]])
sage: g # indirect doctest
[1 1]
[0 1]
sage: g._repr_()
'[1 1]\n[0 1]'
"""
return str(self.matrix())
def _latex_(self):
r"""
EXAMPLES::
sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,0],[0,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: g = G([[1, 1], [0, 1]])
sage: print g._latex_()
\left(\begin{array}{rr}
1 & 1 \\
0 & 1
\end{array}\right)
Type ``view(g._latex_())`` to see the object in an
xdvi window (assuming you have latex and xdvi installed).
"""
return self.matrix()._latex_()
def _act_on_(self, x, self_on_left):
"""
EXAMPLES::
sage: G = GL(4,7)
sage: G.0 * vector([1,2,3,4])
(3, 2, 3, 4)
sage: v = vector(GF(7), [3,2,1,-1])
sage: g = G.1
sage: v * g == v * g.matrix() # indirect doctest
True
"""
if not is_MatrixGroupElement(x) and x not in self.parent().base_ring():
try:
if self_on_left:
return self.matrix() * x
else:
return x * self.matrix()
except TypeError:
return None
def _cmp_(self, other):
"""
EXAMPLES::
sage: F = GF(3); MS = MatrixSpace(F,2)
sage: gens = [MS([1,0, 0,1]), MS([1,1, 0,1])]
sage: G = MatrixGroup(gens)
sage: g = G([1,1, 0,1])
sage: h = G([1,1, 0,1])
sage: g == h
True
sage: g == G.one()
False
"""
return cmp(self.matrix(), other.matrix())
__cmp__ = _cmp_
def list(self):
"""
Return list representation of this matrix.
EXAMPLES::
sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,0],[0,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: g = G.0
sage: g
[1 0]
[0 1]
sage: g.list()
[[1, 0], [0, 1]]
"""
return [list(_) for _ in self.matrix().rows()]
###################################################################
#
# Matrix groups elements implemented with Sage matrices
#
###################################################################
class MatrixGroupElement_generic(MatrixGroupElement_base):
def __init__(self, parent, M, check=True, convert=True):
r"""
Element of a matrix group over a generic ring.
The group elements are implemented as Sage matrices.
INPUT:
- ``M`` -- a matrix.
- ``parent`` -- the parent.
- ``check`` -- bool (default: ``True``). If true does some
type checking.
- ``convert`` -- bool (default: ``True``). If true convert
``M`` to the right matrix space.
TESTS::
sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,0],[0,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: g = G.random_element()
sage: TestSuite(g).run()
"""
if convert:
M = parent.matrix_space()(M)
if check:
if not is_Matrix(M):
raise TypeError('M must be a matrix')
if M.parent() is not parent.matrix_space():
raise TypeError('M must be a in the matrix space of the group')
parent._check_matrix(M)
super(MatrixGroupElement_generic, self).__init__(parent)
if M.is_immutable():
self._matrix = M
else:
self._matrix = M.__copy__()
self._matrix.set_immutable()
def matrix(self):
"""
Obtain the usual matrix (as an element of a matrix space)
associated to this matrix group element.
One reason to compute the associated matrix is that matrices
support a huge range of functionality.
EXAMPLES::
sage: k = GF(7); G = MatrixGroup([matrix(k,2,[1,1,0,1]), matrix(k,2,[1,0,0,2])])
sage: g = G.0
sage: g.matrix()
[1 1]
[0 1]
sage: parent(g.matrix())
Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 7
Matrices have extra functionality that matrix group elements
do not have::
sage: g.matrix().charpoly('t')
t^2 + 5*t + 1
"""
return self._matrix
def _mul_(self,other):
"""
Return the product of self and other, which must have identical
parents.
EXAMPLES::
sage: F = GF(3); MS = MatrixSpace(F,2)
sage: gens = [MS([1,0, 0,1]), MS([1,1, 0,1])]
sage: G = MatrixGroup(gens)
sage: g = G([1,1, 0,1])
sage: h = G([1,1, 0,1])
sage: g*h # indirect doctest
[1 2]
[0 1]
"""
parent = self.parent()
M = self._matrix * other._matrix
# Make it immutable so the constructor doesn't make a copy
M.set_immutable()
return parent.element_class(parent, M, check=False, convert=False)
def __invert__(self):
"""
Return the inverse group element
OUTPUT:
A matrix group element.
EXAMPLES::
sage: G = GL(2,3)
sage: g = G([1,2,1,0]); g
[1 2]
[1 0]
sage: g.__invert__()
[0 1]
[2 1]
sage: g * (~g)
[1 0]
[0 1]
"""
parent = self.parent()
M = ~self._matrix
# Make it immutable so the constructor doesn't make a copy
M.set_immutable()
return parent.element_class(parent, M, check=False, convert=False)
inverse = __invert__
###################################################################
#
# Matrix group elements implemented in GAP
#
###################################################################
class MatrixGroupElement_gap(GroupElementMixinLibGAP, MatrixGroupElement_base, ElementLibGAP):
def __init__(self, parent, M, check=True, convert=True):
r"""
Element of a matrix group over a generic ring.
The group elements are implemented as Sage matrices.
INPUT:
- ``M`` -- a matrix.
- ``parent`` -- the parent.
- ``check`` -- bool (default: ``True``). If true does some
type checking.
- ``convert`` -- bool (default: ``True``). If true convert
``M`` to the right matrix space.
TESTS::
sage: MS = MatrixSpace(GF(3),2,2)
sage: G = MatrixGroup(MS([[1,0],[0,1]]), MS([[1,1],[0,1]]))
sage: G.gen(0)
[1 0]
[0 1]
sage: g = G.random_element()
sage: TestSuite(g).run()
"""
if isinstance(M, GapElement):
ElementLibGAP.__init__(self, parent, M)
return
if convert:
M = parent.matrix_space()(M)
from sage.libs.gap.libgap import libgap
M_gap = libgap(M)
if check:
if not is_Matrix(M):
raise TypeError('M must be a matrix')
if M.parent() is not parent.matrix_space():
raise TypeError('M must be a in the matrix space of the group')
parent._check_matrix(M, M_gap)
ElementLibGAP.__init__(self, parent, M_gap)
def __reduce__(self):
"""
Implement pickling.
TESTS::
sage: MS = MatrixSpace(GF(3), 2, 2)
sage: G = MatrixGroup(MS([[1,0],[0,1]]), MS([[1,1],[0,1]]))
sage: loads(G.gen(0).dumps())
[1 0]
[0 1]
"""
return (self.parent(), (self.matrix(),))
@cached_method
def matrix(self):
"""
Obtain the usual matrix (as an element of a matrix space)
associated to this matrix group element.
EXAMPLES::
sage: F = GF(3); MS = MatrixSpace(F,2,2)
sage: gens = [MS([[1,0],[0,1]]),MS([[1,1],[0,1]])]
sage: G = MatrixGroup(gens)
sage: G.gen(0).matrix()
[1 0]
[0 1]
sage: _.parent()
Full MatrixSpace of 2 by 2 dense matrices over Finite Field of size 3
"""
# We do a slightly specialized version of sage.libs.gap.element.GapElement.matrix()
# in order to use our current matrix space directly and avoid
# some overhead safety checks.
entries = self.gap().Flat()
MS = self.parent().matrix_space()
ring = MS.base_ring()
m = MS([x.sage(ring=ring) for x in entries])
m.set_immutable()
return m