any isometry of Euclidean vector space $\mathbb{R}^3$ has the form
$$\mathbb{R}^3\ni x\mapsto A\cdot x + b\in\mathbb{R}^3,$$
where $A\in O(3)$ is an orthonormal matrix, and $b\in\mathbb{R}^3$. However, geometrically there are only a few types of isometries, namely: translations (iff $A$ is the unit matrix), rotations, reflections, and compositions of those three types of geometric maps.
I am wondering, how to classify those geometric mappings in terms of $A$ and $b$? Are there any properties of $A$ and $b$, such that one can classify the isometries of Euclidean space? I looked in many books, but I could not find anything satisfactory.
I would appreciate your help!
Best Answer
A reflection is an inverse isometry, hence $\det A=-1$, whereas $\det A=1$ for a rotation.
In addition, a rotation has an invariant line, while a reflection has an invariant plane. It follows that, for a rotation, $A$ has only one unit eigenvalue, while for a reflection $A$ has two unit eigenvalues. The corresponding eigenvectors span the rotation axis or the reflection plane.