Suppose I have a matrix $M\in\mathbb{R}^{N\times N}$ such that $M_{ij} = 0$ if $i$ and $j$ are of odd parity (that is to say that if $i$ is even, then $j$ is odd and vice versa). As such, my matrix will look something like this:
$$ \begin{align} \begin{pmatrix} * & 0 & * & 0 & * & \cdots & \\ 0 & * & 0 & * & 0 & \cdots & \\ \vdots \\ \vdots \end{pmatrix}, \end{align} $$
where $*$ is just a numerical place holder. What are these matrices called, and further more are there any interesting properties under certain conditions that have been shown / proven? I'm very interested in its eigenvalues. Unfortunately I have no extra background information to give as I do not know where to start looking.
You can swap rows and columns, such that your matrix becomes $$\begin{pmatrix}*&\ldots&*&0&\ldots&0\\\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\*&\ldots&*&0&\ldots&0\\0&\ldots&0&*&\ldots&*\\\vdots & \ddots & \vdots & \vdots & \ddots & \vdots\\0&\ldots&0&*&\ldots&*\\\end{pmatrix}$$ This is now a so-called block-diagonal matrix. In particular, the eigenvalues of this matrix are the eigenvalues of both blocks. Hence, your eigenvalue problem decomposes into two smaller eigenvalue problems.