Bisymmetric matrix

In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both A = A^T (it is its own transpose), and AJ = JA, where J is the n × n exchange matrix.

For example, any matrix of the form

$${\displaystyle {\begin{bmatrix}a&b&c&d&e\\b&f&g&h&d\\c&g&i&g&c\\d&h&g&f&b\\e&d&c&b&a\end{bmatrix}}={\begin{bmatrix}a_{11}&a_{12}&a_{13}&a_{14}&a_{15}\\a_{12}&a_{22}&a_{23}&a_{24}&a_{14}\\a_{13}&a_{23}&a_{33}&a_{23}&a_{13}\\a_{14}&a_{24}&a_{23}&a_{22}&a_{12}\\a_{15}&a_{14}&a_{13}&a_{12}&a_{11}\end{bmatrix}}}$$

is bisymmetric. The associated ${\displaystyle 5\times 5}$ exchange matrix for this example is

${\displaystyle J_{5}={\begin{bmatrix}0&0&0&0&1\\0&0&0&1&0\\0&0&1&0&0\\0&1&0&0&0\\1&0&0&0&0\end{bmatrix}}}$