Centrosymmetric matrix

This article defines a property that can be evaluated for a square matrix. In other words, given a square matrix (a matrix with an equal number of rows and columns) the matrix either satisfies or does not satisfy the property.
View other properties of square matrices

Definition

Verbal definition

A centrosymmetric matrix is a square matrix that commutes with the exchange matrix (the matrix with 1s on the antidiagonal and 0s elsewhere) of the same number of rows and columns.

Algebraic definition

Suppose $n$ is a positive integer. A $n\times n$ matrix $A=(a_{ij})_{1\leq i\leq n,1\leq j\leq n}$ is termed a centrosymmetric matrix if it satisfies the following condition:

$a_{ij}=a_{n+1-i,n+1-j}\ \forall i,j\in \{1,2,\dots ,\}$

In symbols, if $J$ denotes the exchange matrix, then $A$ is centrosymmetric if and only if $AJ=JA$.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
symmetric Toeplitz matrix				Bisymmetric matrix|FULL LIST, MORE INFO
bisymmetric matrix	centrosymmetric and a symmetric matrix			|FULL LIST, MORE INFO