Bisymmetric matrix

Symmetry pattern of a bisymmetric 5×5 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 and AJ = JA where J is the n × n exchange matrix.

For example:

${\begin{bmatrix}a&b&c&d&e\\b&f&g&h&d\\c&g&i&g&c\\d&h&g&f&b\\e&d&c&b&a\end{bmatrix}}.$

Properties

Bisymmetric matrices are both symmetric centrosymmetric and symmetric persymmetric. It has been shown that 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.

The product of two bisymmetric matrices results in a centrosymmetric matrix.

The inverse of bisymmetric matrices can be represented by recurrence formulas.