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

Ma,b + Mc,d = Ma+c,b+d.

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

-Ma,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

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 Ma,b is invertible and

In other words, any nonzero element Ma,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

Ma, b÷Mc, d = Ma, b×Mc, d-1 = Ma, b×M,

which implies

Ma, b÷Mc, d = M, -

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

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

Ma,b = a M1,0 + b M0,1 = a I2 + b M0,1.

Note that

M0,1 M0,1 = M-1,0 = - I2.

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

a + bi

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

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Author: M.A. Khamsi