How to properly write down the general term of this sequence of matrices?

Question

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.

Answer 1

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.

Answer 2

You can also define $A_n$ as the permutation matrix associated to the permutation $\pi = (1\;2\;3\;\dots\;n)$, written in cycle notation.

But to meet the standard of "crystal clear", we should probably clarify that this is the version of the matrix that acts on a vector $\mathbf x$ by permuting is entries according to $\pi$ (the column representation of $\pi$, as defined on Wikipedia).

Another idea is to define $A_n$ as the matrix with columns $\mathbf e_n, \mathbf e_1, \mathbf e_2, \dots, \mathbf e_{n-1}$, where $\mathbf e_1, \mathbf e_2, \dots, \mathbf e_n$ are the standard basis vectors. (The Wikipedia article linked to above uses this notation for row vectors, but using it for column vectors is more conventional.)

Answer 3

Charlie Johnson also calls these matrices the basic circulants because $$A_n = \text{circ}(0,1,0,\dots,0) = \begin{bmatrix} e_2^\top \\ e_3^\top \\ \vdots \\e_n^\top \\ e_1^\top \end{bmatrix}.$$ Consequently, if $F_n$ denotes the $n$-by-$n$ discrete Fourier transform matrix, then $$A_n = F_n \text{diag}(1,\omega_n,\dots, \omega_n^{n-1})F^*_n,$$ in which $\omega_n := \exp(2\pi\textrm{i}/n)$.