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, any matrix of the form

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

is bisymmetric.

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]

References[]