Consider the following sequence of matrices: $A_n\in\Bbb R^{n\times n}$, defined as
$$A_2=\begin{pmatrix} 0& 1 \\ 1 & 0 \end{pmatrix}, \qquad A_3 = \begin{pmatrix} 0 & 1 & 0\\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}, \qquad A_4 = \begin{pmatrix} 0 & 1 & 0 & 0\\ 0 & 0 & 1& 0 \\ 0 & 0 & 0& 1\\ 1 & 0 & 0 & 0\end{pmatrix}, \qquad \text{etc.}$$
What is a proper, crystal clear, way of writting down the general term $A_n$ of this sequence?
I have tried the following, but am only half satisfied with it and I am wondering if there is smarter/neater way:
$$A_n=\begin{pmatrix} 0 & 1 & 0 &\cdots& 0 \\ \vdots & \ddots & \ddots & \ddots & \vdots\\ \vdots & & \ddots & \ddots &0 \\
0 &\cdots &\cdots & 0 & 1\\ 1 &0 & \cdots & \cdots & 0 \end{pmatrix}, \qquad A_n=\begin{pmatrix} 0 & 1 & 0 &\cdots& 0& 0 \\
0&0&1&\cdots&0& 0\\
%\vdots & \vdots & \vdots & \vdots & \vdots& \vdots\\
\vdots & \vdots & \vdots & & \vdots& \vdots\\
%\vdots & \vdots & \vdots & & \ddots & \vdots\\
%\vdots & & \ddots & \ddots &0 \\
0 &0 & 0 & \cdots & 0 & 1\\ 1 &0 & 0 & \cdots & 0 &0\end{pmatrix}.$$
Do these matrices have a particular standard name? I think they are the adjacency matrices of the cyclic directed path on a graph of $n$ nodes, but I am not sure this fully characterize them.
Disclaimer: I don't think this question belongs to Tex.SE as it about how to best convey a mathematical idea and not about how to type a matrix in latex.
Best Answer
I'd write this (after your first few examples):
In general, $$ a^n_{ij} = \begin{cases}1 & j \equiv i+1 \pmod n \\ 0 & \text{else}\end{cases}, $$ so that $A^n$ is the directed adjacency matrix of a directed cycle of $n$ nodes, or, informally, $A^n$ is $n \times n$, has ones above the diagonal, and in the lower left corner.
Of course, you may choose not to use $A^n$ to indicate the $n$th matrix; I was merely giving an example of a possible notation.
I suppose you could also describe it as a particular Toeplitz matrix, but I'm not sure (depending on context) whether that'd be informative or not. A more compact way would be to say that it's a "circulant matrix on the vector $e_2 \in \Bbb R^n$," but that just sends the reader off to look up "circulant matrix," unless that's an idea that's already in context.