Bisymmetric matrix explained

In mathematics, a bisymmetric matrix is a square matrix that is symmetric about both of its main diagonals. More precisely, an matrix is bisymmetric if it satisfies both (it is its own transpose), and, where is the exchange matrix.

For example, any matrix of the form

$\begina & b & c & d & e \\b & f & g & h & d \\c & g & i & g & c \\d & h & g & f & b \\e & d & c & b & a \end= \begina_ & a_ & a_ & a_ & a_ \\a_ & a_ & a_ & a_ & a_ \\a_ & a_ & a_ & a_ & a_ \\a_ & a_ & a_ & a_ & a_ \\a_ & a_ & a_ & a_ & a_\end$

is bisymmetric. The associated

5 x 5

exchange matrix for this example is

J₅=\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}

Properties

Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric.
The product of two bisymmetric matrices is a centrosymmetric matrix.
Real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues remain the same aside from possible sign changes following pre- or post-multiplication by the exchange matrix.^[1]
If A is a real bisymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be bisymmetric.^[2]
The inverse of bisymmetric matrices can be represented by recurrence formulas.^[3]

Notes and References

Tao . David . Yasuda, Mark . A spectral characterization of generalized real symmetric centrosymmetric and generalized real symmetric skew-centrosymmetric matrices . SIAM Journal on Matrix Analysis and Applications . 23 . 3 . 885–895 . 2002 . 10.1137/S0895479801386730 .
Yasuda . Mark . Some properties of commuting and anti-commuting m-involutions . Acta Mathematica Scientia . 32 . 2 . 631–644 . 2012. 10.1016/S0252-9602(12)60044-7.
Wang. Yanfeng. Lü. Feng. Lü. Weiran. 2018-01-10. The inverse of bisymmetric matrices. Linear and Multilinear Algebra. 67. 3. 479–489. 10.1080/03081087.2017.1422688. 125163794. 0308-1087.