Where can I learn more about shear matrices?


The Wikipedia article is not enough, and sadly it does not have any references.


I understand they are linear transformations. Do they form a group? How do they look like for n-dimensional vectors? How many independent shears can I do in n-dimensions? Are they related to the orthogonal group SO(D), perhaps contained?


Google takes me to this other [website](http://cse.taylor.edu/~btoll/s99/424/res/mtu/Notes/geometry/geo-tran.htm). It is almost enough, but again no references.


I am trying to do some physics with this, so I do not think you will tolerate my explanation. ;-)



Shear transformations
It might be helpful to note that, in two dimensions, a shear transformation is exactly one whose Jordan canonical form is $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ or $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ if you include the identity as a shear transformation. These are exactly the transformations whose eigenvalues are all $1$, so-called unipotent matrices. The unipotent $2 \times 2$ matrices do not form a group: $\begin{pmatrix}1&1\\0&1\end{pmatrix}\times\begin{pmatrix}1&0\\1&1\end{pmatrix}$ is not unipotent.


