Extending an established isomorphism between group rings and a subring of the n × n matrices

In this work, we extend an established isomorphism between group rings and a subring of the [Formula: see text] matrices. This extension allows us to construct more complex matrices over the ring [Formula: see text] We present many interesting examples of complex matrices constructed directly from our extension. We also show that some of the matrices used in the literature before can be obtained by a direct application of our extended isomorphism.