Affine Groups#
AUTHORS:
Volker Braun: initial version
- class sage.groups.affine_gps.affine_group.AffineGroup(degree, ring)#
Bases:
UniqueRepresentation
,Group
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
\[\mathrm{Aff}(A_V) = GL(V) \ltimes V.\]As such, the group elements can be represented by pairs \((A, b)\) of a matrix and a vector. This pair then represents the transformation
\[x \mapsto A x + b.\]We can also represent affine transformations as linear transformations by considering \(\dim(V) + 1\) dimensional space. We take the affine transformation \((A, b)\) to
\[\begin{split}\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix}\end{split}\]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 wherering
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 # needs sage.rings.finite_rings Affine Space of dimension 2 over Finite Field in a of size 2^2 sage: G = AffineGroup(A); G # needs sage.rings.finite_rings Affine Group of degree 2 over Finite Field in a of size 2^2 sage: G is AffineGroup(2,4) # shorthand # needs sage.rings.finite_rings 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:
- Element#
alias of
AffineGroupElement
- cardinality()#
Return the cardinality of
self
.EXAMPLES:
sage: # needs sage.libs.gap sage: AffineGroup(6, GF(5)).cardinality() 172882428468750000000000000000 sage: AffineGroup(6, ZZ).cardinality() +Infinity
- degree()#
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
- linear(A)#
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]
- linear_space()#
Return the space of the affine transformations represented as linear transformations.
We can represent affine transformations \(Ax + b\) as linear transformations by
\[\begin{split}\begin{pmatrix} A & b \\ 0 & 1 \end{pmatrix}\end{split}\]and lifting \(x = (x_1, \ldots, x_n)\) to \((x_1, \ldots, x_n, 1)\).
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
- matrix_space()#
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
- random_element()#
Return a random element of this group.
EXAMPLES:
sage: # needs sage.libs.gap 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
- reflection(v)#
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]
- some_elements()#
Return some elements.
EXAMPLES:
sage: # needs sage.libs.gap sage: G = AffineGroup(4,5) sage: G.some_elements() [ [2 0 0 0] [1] [0 1 0 0] [0] x |-> [0 0 1 0] x + [0] [0 0 0 1] [0], [2 0 0 0] [0] [0 1 0 0] [0] x |-> [0 0 1 0] x + [0] [0 0 0 1] [0], [2 0 0 0] [...] [0 1 0 0] [...] x |-> [0 0 1 0] x + [...] [0 0 0 1] [...]] sage: all(v.parent() is G for v in G.some_elements()) True sage: G = AffineGroup(2,QQ) sage: G.some_elements() [ [1 0] [1] x |-> [0 1] x + [0], ...]
- translation(b)#
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]
- vector_space()#
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