# Centrosymmetric matrix

From Linear

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

## Contents

## 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 is a positive integer. A matrix is termed a **centrosymmetric matrix** if it satisfies the following condition:

In symbols, if denotes the exchange matrix, then is centrosymmetric if and only if .

## 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 | | |