In mathematics, a bisymmetric matrix is a

square matrix In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied. Square matrices are ofte ...

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 :

\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

is bisymmetric.

Properties

*Bisymmetric matrices are both symmetric

centrosymmetric In crystallography, a centrosymmetric point group contains an inversion center as one of its symmetry elements. In such a point group, for every point (x, y, z) in the unit cell there is an indistinguishable point (-x, -y, -z). Such point grou ...

and symmetric persymmetric. *The product of two bisymmetric matrices is a centrosymmetric matrix. *

Real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (201 ...

-valued bisymmetric matrices are precisely those symmetric matrices whose

eigenvalue In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denot ...

s 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 Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...

of bisymmetric matrices can be represented by recurrence formulas.

References

{{DEFAULTSORT:Bisymmetric Matrix Matrices