Bisymmetric matrix

From Wikipedia, the free encyclopedia
Jump to: navigation, search

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:

\begin{bmatrix}
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{bmatrix}.

Properties[edit]

Bisymmetric matrices are both symmetric centrosymmetric 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.[1]

The product of two bisymmetric matrices results in a centrosymmetric matrix

References[edit]