Bisymmetric Matrix

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

$\begin a & b & c & d & e \\ b & f & g & h & d \\ c & g & i & g & c \\ d & h & g & f & b \\ e & d & c & b & a \end = \begin a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \\ a_ & a_ & a_ & a_ & a_ \end$

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

$J_ = \begin 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \end$

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

Matrices (mathematics)

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