# Affine transformation matrix coefficients

In an affine transformation $x \mapsto Ax+b$, $b$ represents the translation; but what does the matrix $A$ represent exactly? In a 2D example, $A$ is a $2\times 2$ matrix, but what does each term represent?

Suppose that given an affine transformation $\operatorname{Aff}_1$ I want to introduce another affine transformation $\operatorname{Aff}_2$ that is "half" the transformation $\operatorname{Aff}_1$. For the translation $b$, I just need to divide each coefficient by $2$, but what about the matrix $A$?

For $A \in GL(2, \mathbb{R})$, the map $x \mapsto Ax$ is an invertible linear transformation from $\mathbb{R}^2$ to itself. There are four types of such transformations:
Furthermore, the translation by $b$ complicates matters. Let $f(x) = Ax + b$ be a given affine transformation and suppose $g(x) = Cx + d$ is the affine transformation which 'does half' of $f(x)$. What does this mean? Well, I'm going to take it to mean that if you do the affine transformation $g(x)$ twice, then you get $f(x)$. So we are trying to find $C \in GL(2, \mathbb{R})$ and $d \in \mathbb{R}^2$ such that $$Ax + b = f(x) = g(g(x)) = g(Cx + d) = C(Cx + d) + d = C^2x + (Cd + d).$$ That is, we want to find $C$ and $d$ such that $A = C^2$ and $b = Cd + d = (C+I)d$. Depending on what $A$ and $b$ are, there may be no such $C$ and $d$ (as is the case when $A$ defines a reflection).