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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 $2^k$.)

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