# Bisymmetric 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

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

1. It is both a symmetric matrix and a centrosymmetric matrix.
2. It is both a symmetric matrix and a persymmetric matrix.
3. 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 |