TITLE: Do matrices with only elements along the main and anti-diagonals have a name?
QUESTION [5 upvotes]: To expand upon the title, I am wondering if there is a specific name for square matrices of the form:  $$M = \begin{bmatrix} a_{11} & 0 & \cdots & 0 & \cdots & & 0 & b_{1n} \\ 0 & a_{22} & 0 & \cdots & & b_{2 \ i} & & 0 \\ 0 & \ddots & \ddots & & \cdot^{\textstyle \cdot^{\textstyle \cdot}} & 0 & & 0 \\ & & & a_{} &  & \vdots  &  & \\ \vdots & 0 & \cdot^{\textstyle \cdot^{\textstyle \cdot}} & 0 & \ddots & 0 & & \vdots\\ & \cdots & b_{i \ n-(i+1)} & \cdots & & a_{i \ i} &\\ 0 & \cdot^{{\textstyle \cdot}^{\textstyle \cdot}} & & &  &  \ddots & \ddots & 0\\ b_{n 1} & 0 & \cdots & 0 & & \cdots & & a_{nn} \end{bmatrix} \\$$ 
where, $ M = A_{\text{diagonal}} + B_{\text{anti-diagonal}}$
Essentially, M is a square matrix which is zero everywhere except the main and anti-diagonals. In other words, it is the sum of diagonal and anti-diagonal square matrices with the same dimension.
Apologies for the formatting, the normal LaTeX notation for up-right dots is not supported on this site.

REPLY [6 votes]: I am continuing in the answer box, to get this out of the "unanswered" queue. The OP asks "for a more standard terminology that is perhaps present in the literature." 
The name "X-matrices" or "X-form matrices" has also been used in the published literature, for example, Properties of Central Symmetric X-Form Matrices (2011) and The exponential functions of central-symmetric X-form matrices (2016).
The reason, I guess, why this name is not used more extensively, is that it seems more natural to reorder the basis vectors and work with $2\times 2$ block-diagonal matrices.