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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.

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    $\begingroup$ they have been called "X-matrices" in this post; conjugation with a permutation matrix brings them into a block-diagonal form (2x2 blocks along the main diagonal), so you could just work in that representation. $\endgroup$ – Carlo Beenakker Apr 3 at 16:37
  • $\begingroup$ @CarloBeenakker Is every real (complex) matrix orthonormal (unitary) equivalent to a x matrix? $\endgroup$ – Ali Taghavi Apr 4 at 19:25
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    $\begingroup$ @AliTaghavi -- I don't think so, because not every matrix has a complete set of two-dimensional invariant subspaces, for example $\tiny\left( \begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ \end{array} \right)$ $\endgroup$ – Carlo Beenakker Apr 5 at 8:00
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    $\begingroup$ +1 for the rising dots. $\endgroup$ – Wolfgang Apr 7 at 5:56
  • $\begingroup$ @CarloBeenakker Thank you for your comment. I was not aware of the post referring to them as X matrices which is a very fitting name; however, I was hoping for a more standard terminology that is perhaps present in the literature. I will keep looking and report back at a later date. $\endgroup$ – Victoria M Apr 7 at 14:42
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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.

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    $\begingroup$ Thank you for looking into this :) . I'll mark this as the answer since I was looking for the terminology in current literature. $\endgroup$ – Victoria M Apr 7 at 15:34

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