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

[ 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 ] = [ 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 ] {\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 5 × 5 {\displaystyle 5\times 5} exchange matrix for this example is

J 5 = [ 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 ] {\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}}}

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