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
is bisymmetric. The associated exchange matrix for this example is
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.
References