Bisymmetric matrix

Short description: Square matrix symmetric about both its diagonal and anti-diagonal

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 (it is its own transpose), and AJ = JA, where J is the n × n exchange matrix.

For example, any matrix of the form

$$\left[\begin{array}{ccccc}a& b& c& d& e\\ b& f& g& h& d\\ c& g& i& g& c\\ d& h& g& f& b\\ e& d& c& b& a\end{array}\right]=\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& a_{14}& a_{15}\\ a_{12}& a_{22}& a_{23}& a_{24}& a_{14}\\ a_{13}& a_{23}& a_{33}& a_{23}& a_{13}\\ a_{14}& a_{24}& a_{23}& a_{22}& a_{12}\\ a_{15}& a_{14}& a_{13}& a_{12}& a_{11}\end{array}\right]$$

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

$J_5=\left[\begin{array}{ccccc}0& 0& 0& 0& 1\\ 0& 0& 0& 1& 0\\ 0& 0& 1& 0& 0\\ 0& 1& 0& 0& 0\\ 1& 0& 0& 0& 0\end{array}\right]$

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.
• If A is a real bisymmetric matrix with distinct eigenvalues, then the matrices that commute with A must be bisymmetric.
• The inverse of bisymmetric matrices can be represented by recurrence formulas.