Matrix Representation of the Multiplicative Group of Complex Numbers

The set of all nonzero complex numbers forms a group under complex multiplication; that is, it meets the requirement of closure, existence of identity and inverses, and associativity. There is a bijection between and a set of real matrices that respects the multiplicative structure of both sets, i.e., , where the multiplication on the left is of nonzero complex numbers, and the multiplication on the right is of matrices. This bijection is given by . For any complex number , is the matrix inverse of .