Matrix Representation of the Multiplicative Group of Complex Numbers

first number

real part

1

imaginary part

2

second number

real part

3

imaginary part

4

(1 + 2 i) (3 + 4 i) = -5 + 10 i



1	-2
2	1

 

3	-4
4	3

 = 

-5	-10
10	-5



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

ϕ

between

*



and a set of

2×2

real matrices that respects the multiplicative structure of both sets, i.e.,

ϕ(zw)=ϕ(z)ϕ(w)

, 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

ϕ(a+bi)=

a	-b
b	a



. For any complex number

z

,

ϕ(1/z)

is the matrix inverse of

ϕ(z)

.