Elements of Affine Groups#
The class in this module is used to represent the elements of
AffineGroup()
and its
subgroups.
EXAMPLES:
sage: F = AffineGroup(3, QQ)
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: G = AffineGroup(2, ZZ)
sage: g = G([[1,1],[0,1]], [1,0])
sage: h = G([[1,2],[0,1]], [0,1])
sage: g*h
[1 3] [2]
x |-> [0 1] x + [1]
sage: h*g
[1 3] [1]
x |-> [0 1] x + [1]
sage: g*h != h*g
True
AUTHORS:
Volker Braun
- class sage.groups.affine_gps.group_element.AffineGroupElement(parent, A, b=0, convert=True, check=True)#
Bases:
MultiplicativeGroupElement
An affine group element.
INPUT:
A
– an invertible matrix, or something defining a matrix ifconvert==True
.b
– a vector, or something defining a vector ifconvert==True
(default:0
, defining the zero vector).parent
– the parent affine group.convert
- bool (default:True
). Whether to convertA
into the correct matrix space andb
into the correct vector space.check
- bool (default:True
). Whether to do some checks or just accept the input as valid.
As a special case,
A
can be a matrix obtained frommatrix()
, that is, one row and one column larger. In that case, the group element defining that matrix is reconstructed.OUTPUT: The affine group element \(x \mapsto Ax + b\)
EXAMPLES:
sage: G = AffineGroup(2, GF(3)) sage: # needs sage.libs.gap sage: g = G.random_element() sage: type(g) <class 'sage.groups.affine_gps.affine_group.AffineGroup_with_category.element_class'> sage: G(g.matrix()) == g True sage: G(2) [2 0] [0] x |-> [0 2] x + [0]
Conversion from a matrix and a matrix group element:
sage: M = Matrix(4, 4, [0, 0, -1, 1, 0, -1, 0, 1, -1, 0, 0, 1, 0, 0, 0, 1]) sage: A = AffineGroup(3, ZZ) sage: A(M) [ 0 0 -1] [1] x |-> [ 0 -1 0] x + [1] [-1 0 0] [1] sage: G = MatrixGroup([M]) sage: A(G.0) [ 0 0 -1] [1] x |-> [ 0 -1 0] x + [1] [-1 0 0] [1]
- A()#
Return the general linear part of an affine group element.
OUTPUT: The matrix \(A\) of the affine group element \(Ax + b\).
EXAMPLES:
sage: G = AffineGroup(3, QQ) sage: g = G([1,2,3,4,5,6,7,8,0], [10,11,12]) sage: g.A() [1 2 3] [4 5 6] [7 8 0]
- b()#
Return the translation part of an affine group element.
OUTPUT: The vector \(b\) of the affine group element \(Ax + b\).
EXAMPLES:
sage: G = AffineGroup(3, QQ) sage: g = G([1,2,3,4,5,6,7,8,0], [10,11,12]) sage: g.b() (10, 11, 12)
- list()#
Return list representation of
self
.EXAMPLES:
sage: F = AffineGroup(3, QQ) sage: g = F([1,2,3,4,5,6,7,8,0], [10,11,12]) sage: g [1 2 3] [10] x |-> [4 5 6] x + [11] [7 8 0] [12] sage: g.matrix() [ 1 2 3|10] [ 4 5 6|11] [ 7 8 0|12] [--------+--] [ 0 0 0| 1] sage: g.list() [[1, 2, 3, 10], [4, 5, 6, 11], [7, 8, 0, 12], [0, 0, 0, 1]]
- matrix()#
Return the standard matrix representation of
self
.See also
EXAMPLES:
sage: G = AffineGroup(3, GF(7)) sage: g = G([1,2,3,4,5,6,7,8,0], [10,11,12]) sage: g [1 2 3] [3] x |-> [4 5 6] x + [4] [0 1 0] [5] sage: g.matrix() [1 2 3|3] [4 5 6|4] [0 1 0|5] [-----+-] [0 0 0|1] sage: parent(g.matrix()) Full MatrixSpace of 4 by 4 dense matrices over Finite Field of size 7 sage: g.matrix() == matrix(g) True
Composition of affine group elements equals multiplication of the matrices:
sage: # needs sage.libs.gap sage: g1 = G.random_element() sage: g2 = G.random_element() sage: g1.matrix() * g2.matrix() == (g1*g2).matrix() True