Is it possible to write a C/C++, python, matlab, mathematica or maple function, for any 4$\times$4 input real homogeneous matrix, to determine what kind of geometric transformation it is?
Take the following shear matrix as an example $$ S=\left( \begin{array}{cccc} 1 & t & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)$$ The Jordan normal form (decomposition) of which is:
$$A=\left( \begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & \frac{1}{t} \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ \end{array} \right)$$
and $$B=\left( \begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)$$
where: $$S=A\cdot B\cdot A^{-1}$$
while the $B$ obtained here is usually considered as a translation. How does Jordan normal form work to identify the original $S$ as shear?
This is not an answer but might provide clue in finally resolving you problem.
First, there are actually no widely accepted analytical definitions for the geometrical transformations in homogeneous form as you have mentioned. For example, in the most recent book I can see on this topic, Chapter 9 of which is on ** Geometric Transforms**.
(John Vince, Mathematics for Computer Graphics, Fifth Edition, Springer-Verlag London Ltd., 2017 (ISBN 978-1-4471-7334-2 ISBN 978-1-4471-7336-6 (eBook) DOI 10.1007/978-1-4471-7336-6))
But I didn't see any valuable information that can be used for your question.
Additionally, Multiple View Geometry in Computer Vision,Second Edition, Cambridge University Press, 2003 by Prof. Richard Hartley at Australian National University, Canberra, Australia and Prof. Andrew Zisserman at University of Oxford, UK, has the reputation of The Bible in multiple view geometry. As a matter of fact, there is not immediate answer to your question yet.
However, in this link https://arxiv.org/abs/1307.0998, there is a paper uploaded to arXiv.org database several years ago, where you may be able to find something more useful and might be suitable for your qeustion. The concepts and ideas in this article looks just conventional; the only difference might be:
this is the only analytical definitions on the homogenous matrices of common geometrical transformations, when trying to solve the problem you asked here you would understand why I say it is the only analytical definition;
several different kind of geometrical transformations under this viewpoint, have actually the same geometrical essence, which can be represented in the same the so-called Householder-Chen elementary matrix.
Though still it didnot give an immediate answer for all arbitrary 4$\times$4 matrices, since Householder elementary matrices are basic factors of any other matrices, it seems your answer should be in it.
Hope this help.