Given two sets of corresponding points $\mathbf{P}_1, \mathbf{P}_2 \in \mathbb{R}^{d \times n}$, where $\mathbf{P}_1, \mathbf{P}_2$ are pointclouds expressed as matrices, we can derive a matrix $\mathbf{A} \in \mathbb{R}^{d \times d}$ such that:
\begin{equation} \mathbf{A} = \arg \min_{\mathbf{A}} \sum_{i=1}^n \| \mathbf{A}\mathbf{P}_{2,i} - \mathbf{P}_{1,i} \|_1 \end{equation}
where $\mathbf{P}_{1,i}$ is the $i$-th point of $\mathbf{P}_{1}$ and $\|\cdot\|_1$ is the $1$-norm.
If $\mathbf{P}_1 \mapsto \mathbf{P}_2$ denotes something resembling a non-affine transformation (e.g.: isometric bending), the matrix $\mathbf{A}$ closely approximates such non-affine transformations.
My question is a bit generic: I want to know more about the properties and possible decomposition of such general $\mathbb{R}^{d \times d}$ matrices, I care most about $d=3$. They are neither affine, nor projective. I could not find a lot of literature about them. Please point me to the right direction.