Example of a 3x3 Centrosymmetric Matrix

General Form

A 3x3 centrosymmetric matrix follows the pattern where elements are symmetric about the geometric center of the matrix. The general form is:

Concrete Example

By assigning arbitrary values (e.g., ), we obtain the following matrix:

Verification

Verification via Index Symmetry

For a matrix, the centrosymmetric condition is . We can check the corresponding pairs:

• Corners:

  • and

  • and

• Edges:

  • and

  • and

• Center:

  • (matches itself)

Verification via Exchange Matrix Commutation

A matrix is centrosymmetric if and only if it commutes with the exchange matrix . For , the exchange matrix is:

Performing the multiplication :

Performing the multiplication :

Since , the matrix is confirmed to be centrosymmetric.