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.



Do matrices with only elements along the main and anti-diagonals have a name?
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](https://www.researchgate.net/publication/236874314_Properties_of_Central_Symmetric_X-Form_Matrices) (2011) and [The exponential functions of central-symmetric X-form matrices](https://jmm.guilan.ac.ir/article_1804.html) (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.


