# Bisymmetric 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

A **bisymmetric matrix** is a square matrix that satisfies the following equivalent conditions:

- It is both a symmetric matrix and a centrosymmetric matrix.
- It is both a symmetric matrix and a persymmetric matrix.
- It is both a centrosymmetric matrix and a persymmetric matrix.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

symmetric Toeplitz matrix | | | |||

scalar matrix | | |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

symmetric matrix | symmetric about the main diagonal (top left to bottom right), or, equal to its matrix transpose | | | ||

persymmetric matrix | symmetric about the antidiagonal (bottom left to top right), or, commutes with the exchange matrix | | | ||

centrosymmetric matrix | symmetric about the center point. Note that this definition, although typically used for square matrices, also makes sense for rectangulra matrices | | |