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r""" Affine Groups
AUTHORS:
- Volker Braun: initial version """
############################################################################## # Copyright (C) 2013 Volker Braun <vbraun.name@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ ##############################################################################
#################################################################
r""" An affine group.
The affine group `\mathrm{Aff}(A)` (or general affine group) of an affine space `A` is the group of all invertible affine transformations from the space into itself.
If we let `A_V` be the affine space of a vector space `V` (essentially, forgetting what is the origin) then the affine group `\mathrm{Aff}(A_V)` is the group generated by the general linear group `GL(V)` together with the translations. Recall that the group of translations acting on `A_V` is just `V` itself. The general linear and translation subgroups do not quite commute, and in fact generate the semidirect product
.. MATH::
\mathrm{Aff}(A_V) = GL(V) \ltimes V.
As such, the group elements can be represented by pairs `(A, b)` of a matrix and a vector. This pair then represents the transformation
.. MATH::
x \mapsto A x + b.
We can also represent affine transformations as linear transformations by considering `\dim(V) + 1` dimensonal space. We take the affine transformation `(A, b)` to
.. MATH::
\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix}
and lifting `x = (x_1, \ldots, x_n)` to `(x_1, \ldots, x_n, 1)`. Here the `(n + 1)`-th component is always 1, so the linear representations acts on the affine hyperplane `x_{n+1} = 1` as affine transformations which can be seen directly from the matrix multiplication.
INPUT:
Something that defines an affine space. For example
- An affine space itself:
* ``A`` -- affine space
- A vector space:
* ``V`` -- a vector space
- Degree and base ring:
* ``degree`` -- An integer. The degree of the affine group, that is, the dimension of the affine space the group is acting on.
* ``ring`` -- A ring or an integer. The base ring of the affine space. If an integer is given, it must be a prime power and the corresponding finite field is constructed.
* ``var`` -- (default: ``'a'``) Keyword argument to specify the finite field generator name in the case where ``ring`` is a prime power.
EXAMPLES::
sage: F = AffineGroup(3, QQ); F Affine Group of degree 3 over Rational Field sage: F(matrix(QQ,[[1,2,3],[4,5,6],[7,8,0]]), vector(QQ,[10,11,12])) [1 2 3] [10] x |-> [4 5 6] x + [11] [7 8 0] [12] sage: F([[1,2,3],[4,5,6],[7,8,0]], [10,11,12]) [1 2 3] [10] x |-> [4 5 6] x + [11] [7 8 0] [12] sage: F([1,2,3,4,5,6,7,8,0], [10,11,12]) [1 2 3] [10] x |-> [4 5 6] x + [11] [7 8 0] [12]
Instead of specifying the complete matrix/vector information, you can also create special group elements::
sage: F.linear([1,2,3,4,5,6,7,8,0]) [1 2 3] [0] x |-> [4 5 6] x + [0] [7 8 0] [0] sage: F.translation([1,2,3]) [1 0 0] [1] x |-> [0 1 0] x + [2] [0 0 1] [3]
Some additional ways to create affine groups::
sage: A = AffineSpace(2, GF(4,'a')); A Affine Space of dimension 2 over Finite Field in a of size 2^2 sage: G = AffineGroup(A); G Affine Group of degree 2 over Finite Field in a of size 2^2 sage: G is AffineGroup(2,4) # shorthand True
sage: V = ZZ^3; V Ambient free module of rank 3 over the principal ideal domain Integer Ring sage: AffineGroup(V) Affine Group of degree 3 over Integer Ring
REFERENCES:
- :wikipedia:`Affine_group` """ def __classcall__(cls, *args, **kwds): """ Normalize input to ensure a unique representation.
EXAMPLES::
sage: A = AffineSpace(2, GF(4,'a')) sage: AffineGroup(A) is AffineGroup(2,4) True sage: AffineGroup(A) is AffineGroup(2, GF(4,'a')) True sage: A = AffineGroup(2, QQ) sage: V = QQ^2 sage: A is AffineGroup(V) True """ return V
""" Initialize ``self``.
INPUT:
- ``degree`` -- integer. The degree of the affine group, that is, the dimension of the affine space the group is acting on naturally.
- ``ring`` -- a ring. The base ring of the affine space.
EXAMPLES::
sage: Aff6 = AffineGroup(6, QQ) sage: Aff6 == Aff6 True sage: Aff6 != Aff6 False
TESTS::
sage: G = AffineGroup(2, GF(5)); G Affine Group of degree 2 over Finite Field of size 5 sage: TestSuite(G).run() """
""" Verify that ``A``, ``b`` define an affine group element and raises a ``TypeError`` if the input does not define a valid group element.
This is called from the group element constructor and can be overridden for subgroups of the affine group. It is guaranteed that ``A``, ``b`` are in the correct matrix/vector space.
INPUT:
- ``A`` -- an element of :meth:`matrix_space`
- ``b`` -- an element of :meth:`vector_space`
TESTS::
sage: Aff3 = AffineGroup(3, QQ) sage: A = Aff3.matrix_space()([1,2,3,4,5,6,7,8,0]) sage: det(A) 27 sage: b = Aff3.vector_space().an_element() sage: Aff3._element_constructor_check(A, b)
sage: A = Aff3.matrix_space()([1,2,3,4,5,6,7,8,9]) sage: det(A) 0 sage: Aff3._element_constructor_check(A, b) Traceback (most recent call last): ... TypeError: A must be invertible """
r""" Return a LaTeX representation of ``self``.
EXAMPLES::
sage: G = AffineGroup(6, GF(5)) sage: latex(G) \mathrm{Aff}_{6}(\Bold{F}_{5}) """
""" Return a string representation of ``self``.
EXAMPLES::
sage: AffineGroup(6, GF(5)) Affine Group of degree 6 over Finite Field of size 5 """
""" Return the dimension of the affine space.
OUTPUT:
An integer.
EXAMPLES::
sage: G = AffineGroup(6, GF(5)) sage: g = G.an_element() sage: G.degree() 6 sage: G.degree() == g.A().nrows() == g.A().ncols() == g.b().degree() True """
def matrix_space(self): """ Return the space of matrices representing the general linear transformations.
OUTPUT:
The parent of the matrices `A` defining the affine group element `Ax+b`.
EXAMPLES::
sage: G = AffineGroup(3, GF(5)) sage: G.matrix_space() Full MatrixSpace of 3 by 3 dense matrices over Finite Field of size 5 """
def vector_space(self): """ Return the vector space of the underlying affine space.
EXAMPLES::
sage: G = AffineGroup(3, GF(5)) sage: G.vector_space() Vector space of dimension 3 over Finite Field of size 5 """
def linear_space(self): r""" Return the space of the affine transformations represented as linear transformations.
We can represent affine transformations `Ax + b` as linear transformations by
.. MATH::
\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix}
and lifting `x = (x_1, \ldots, x_n)` to `(x_1, \ldots, x_n, 1)`.
.. SEEALSO::
- :meth:`sage.groups.affine_gps.group_element.AffineGroupElement.matrix()`
EXAMPLES::
sage: G = AffineGroup(3, GF(5)) sage: G.linear_space() Full MatrixSpace of 4 by 4 dense matrices over Finite Field of size 5 """
""" Construct the general linear transformation by ``A``.
INPUT:
- ``A`` -- anything that determines a matrix
OUTPUT:
The affine group element `x \mapsto A x`.
EXAMPLES::
sage: G = AffineGroup(3, GF(5)) sage: G.linear([1,2,3,4,5,6,7,8,0]) [1 2 3] [0] x |-> [4 0 1] x + [0] [2 3 0] [0] """
""" Construct the translation by ``b``.
INPUT:
- ``b`` -- anything that determines a vector
OUTPUT:
The affine group element `x \mapsto x + b`.
EXAMPLES::
sage: G = AffineGroup(3, GF(5)) sage: G.translation([1,4,8]) [1 0 0] [1] x |-> [0 1 0] x + [4] [0 0 1] [3] """
""" Construct the Householder reflection.
A Householder reflection (transformation) is the affine transformation corresponding to an elementary reflection at the hyperplane perpendicular to `v`.
INPUT:
- ``v`` -- a vector, or something that determines a vector.
OUTPUT:
The affine group element that is just the Householder transformation (a.k.a. Householder reflection, elementary reflection) at the hyperplane perpendicular to `v`.
EXAMPLES::
sage: G = AffineGroup(3, QQ) sage: G.reflection([1,0,0]) [-1 0 0] [0] x |-> [ 0 1 0] x + [0] [ 0 0 1] [0] sage: G.reflection([3,4,-5]) [ 16/25 -12/25 3/5] [0] x |-> [-12/25 9/25 4/5] x + [0] [ 3/5 4/5 0] [0] """ except ZeroDivisionError: raise ValueError('v has norm zero')
""" Return a random element of this group.
EXAMPLES::
sage: G = AffineGroup(4, GF(3)) sage: G.random_element() # random [2 0 1 2] [1] [2 1 1 2] [2] x |-> [1 0 2 2] x + [2] [1 1 1 1] [2] sage: G.random_element() in G True """
def _an_element_(self): """ Return an element.
TESTS::
sage: G = AffineGroup(4,5) sage: G.an_element() in G True """
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