# Is there a name for this sort of mod(n)-Diagonal Matrix?

I've been tinkering around with some linear algebra lately and have stumbled across some problems where matrices keep popping up (even for matrices of larger sizes) with the form: $$X =\begin{pmatrix} x_{11} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & x_{23} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & x_{35} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & x_{47} & 0\\ x_{51} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & x_{63} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & x_{75} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & x_{87} & 0\\ \end{pmatrix}$$ For a typical diagonal matrix $$D$$, the entries occur along a diagonal such that $$d_{i,j}\neq0$$ only when $$j=i$$ (for the main diagonal), or something like $$d_{i,j}\neq0$$ only when $$j=i\pm k$$ (for diagonals $$\pm k$$ steps above the main diagonal.)

In my case the matrices have nonzeros only along the "diagonal" where $$x_{i,j}\neq0$$ only when $$j=mod_{n}(2i-2)+1$$ where $$n$$ is the dimension of the square matrix (or it gets a bit cleaner if you index it starting from $$x_{00}$$ like a programmer would).

Is there a conventional name for a matrix like this? I'm hoping that there might be more information about their properties in literature, especially with regards to the forms of the eigenvalues (when $$n$$ gets large enough that it's not a simple quadratic or something).

(Also, as I'm not sure whether it is relevant or not, all the matrices I'm looking at are square matrices of side length $$2^k$$.)