Affine Group - Matrix Representation

Matrix Representation

Representing the affine group as a semidirect product of V by GL(V), then by construction of the semidirect product, the elements are pairs (M, v), where v is a vector in V and M is a linear transform in GL(V), and multiplication is given by:

This can be represented as the (n + 1)×(n + 1) block matrix:

where M is an n×n matrix over K, v an n × 1 column vector, 0 is a 1 × n row of zeros, and 1 is the 1 × 1 identity block matrix.

Formally, Aff(V) is naturally isomorphic to a subgroup of, with V embedded as the affine plane, namely the stabilizer of this affine plane; the above matrix formulation is the (transpose of) the realization of this, with the (n × n and 1 × 1) blocks corresponding to the direct sum decomposition .

A similar representation is any (n + 1)×(n + 1) matrix in which the entries in each column sum to 1. The similarity P for passing from the above kind to this kind is the (n + 1)×(n + 1) identity matrix with the bottom row replaced by a row of all ones.

Each of these two classes of matrices is closed under matrix multiplication.