Centrosymmetric Group

Consider the set of $\text{[math]}$ matrices that satisfy the property

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

Let

The matrix $\text{[math]}$ 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 $\text{[math]}$ forms a group.

Proof. It can be easily shown that the identity is in the group, i.e. $\text{[math]}$ . Also the set is closed under multiplication, i.e. if $\text{[math]}$ and $\text{[math]}$ then $\text{[math]}$ . To see why, consider

If $\text{[math]}$ , then $\text{[math]}$

Therefore, $\text{[math]}$ . This completes the proof that $\text{[math]}$ forms a group.