mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 often ...

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 In mathematics, especially linear algebra, the exchange matrices (also called the reversal matrix, backward identity, or standard involutory permutation) are special cases of permutation matrix, permutation matrices, where the 1 elements reside on t ...

. 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 g ...

and symmetric

persymmetric In mathematics, persymmetric matrix may refer to: # a square matrix which is symmetric with respect to the northeast-to-southwest diagonal; or # a square matrix such that the values on each line perpendicular to the main diagonal are the same for a ...

. *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) (2010) ...

-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 denoted b ...

s remain the same aside from possible sign changes following pre- or post-multiplication by the

. *If ''A'' is a real bisymmetric matrix with distinct eigenvalues, then the matrices that

commute Commute, commutation or commutative may refer to: * Commuting, the process of travelling between a place of residence and a place of work Mathematics * Commutative property, a property of a mathematical operation whose result is insensitive to th ...

with ''A'' must be bisymmetric. *The inverse of bisymmetric matrices can be represented by recurrence formulas.

References

{{DEFAULTSORT:Bisymmetric Matrix Matrices