Bisymmetric matrix
Encyclopedia
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 = AT and AJ = JA where J is the n × n exchange matrix
.
For example:
and symmetric persymmetric
. It has been shown that real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues are the same up to sign after pre or post multiplication by the exchange matrix.
The product of two bisymmetric matrices results in a centrosymetric matrix
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a bisymmetric matrix is a square matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
that is symmetric about both of its main diagonals. More precisely, an n × n matrix A is bisymmetric if it satisfies both A = AT and AJ = JA where J is the n × n exchange matrix
Exchange matrix
In mathematics, especially linear algebra, the exchange matrix is a special case of a permutation matrix, where the 1 elements reside on the counterdiagonal and all other elements are zero...
.
For example:
Properties
Bisymmetric matrices are both symmetric centrosymmetricCentrosymmetric matrix
In mathematics, especially in linear algebra and matrix theory, a centrosymmetric matrix is a matrix which is symmetric about its center. More precisely, an n × n matrix A = [ Ai,j ] is centrosymmetric when its entries satisfy...
and symmetric persymmetric
Persymmetric matrix
In mathematics, persymmetric matrix may refer to:# a square matrix which is symmetric in 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 given line....
. It has been shown that real-valued bisymmetric matrices are precisely those symmetric matrices whose eigenvalues are the same up to sign after pre or post multiplication by the exchange matrix.
The product of two bisymmetric matrices results in a centrosymetric matrix