Complex numbers as Matrices

Complex numbers as Matrices

In this section, we use matrices to give a representation of complex numbers. Indeed, consider the set

We will write

Clearly, the set is not empty. For example, we have

In particular, we have

for any real numbers a, b, c, and d.

Algebraic Properties of

1.

Addition: For any real numbers a, b, c, and d, we have

M_a,b + M_c,d = M_a+c,b+d.

In other words, if we add two elements of the set , we still get a matrix in . In particular, we have

-M_a,b = M_-a,-b.

 

2.

Multiplication by a number: We have

So a multiplication of an element of and a number gives a matrix in .

2.

Multiplication: For any real numbers a, b, c, and d, we have

In other words, we have

M_a,b M_c,d = M_{ac-bd, ad+bc}.

This is an extraordinary formula. It is quite conceivable given the difficult form of the matrix multiplication that, a priori, the product of two elements of may not be in again. But, in this case, it turns out to be true.

The above properties infer to a very nice structure. The next natural question to ask, in this case, is whether a nonzero element of is invertible. Indeed, for any real numbers a and b, we have

So, if , the matrix M_a,b is invertible and

In other words, any nonzero element M_a,b of is invertible and its inverse is still in since

In order to define the division in , we will use the inverse. Indeed, recall that

So for the set , we have

M_{a, b}�M_{c, d} = M_{a, b}�M_{c, d}^-1 = M_{a, b}�M_,

which implies

M_{a, b}�M_{c, d} = M_{, -}  

The matrix M_a,-b is called the conjugate of M_a,b. Note that the conjugate of the conjugate of M_a,b is M_a,b itself.

Fundamental Equation. For any M_a,b in , we have

Note that

Remark. If we introduce an imaginary number i such that i² = -1, then the matrix M_a,b may be rewritten by

A lot can be said about , but we will advise you to visit the page on complex numbers.