Centrosymmetric GroupΒΆ

Consider the set of matrices that satisfy the property

Explicitly, in matrix form, the elements are arranged as follows


The matrix is sometimes referred to as the exchange matrix.

Using the exchange matrix the above set of matrices can also be conveniently defined as

Proposition. The set of matrices forms a group.

Proof. It can be easily shown that the identity is in the group, i.e. . Also the set is closed under multiplication, i.e. if and then . To see why, consider

If , then

Therefore, . This completes the proof that forms a group.