In this explainer, we will learn how to represent a complex number as a linear transformation matrix and use this to determine the product of two complex numbers.
When we first start learning about matrices, we often make connections to operations in the more familiar real numbers to help grasp the new concepts. We soon discover that there are lots of properties of matrices that do not have analogues in the real numbers. For example, matrix multiplication is, in general, noncommutative. That is, for two matrices and , in general. Nevertheless, there are helpful analogues with the real numbers that give us insightful intuition about matrices. One of the most persistent analogues with the real numbers is the correspondence between the identity matrix and the multiplicative identity in the real numbers: 1.
Many of the familiar properties of the multiplicative identity carry over directly to matrix algebra; for example, , and . In fact, there is a direct correspondence between the set of all matrices of the form for a constant and the real numbers, so much so that, mathematically, we can consider them to be equivalent or, to be precise, isomorphic. However, even when it comes to the analogue between the identity matrix and multiplicative identity in the reals, there are differences. In this explainer, we will explore the implications of one such difference in the case of 2-by-2 matrices.
We begin by considering the square of the matrix
Using matrix multiplication, we have
Interestingly, this is equal to the negative identity matrix. Hence, . What we have shown is that there exists a matrix whose square is negative the identity. This is clearly a place where the correspondence with the real numbers does not hold since there is no real number such that . However, there is a complex number which has this property. This leads us to a question of whether we can find some correspondence between complex numbers and 2-by-2 matrices. Since we can consider the matrix to correspond to , it makes sense that the imaginary numbers would correspond to constant multiples of . Furthermore, since there is a direct correspondence between the real numbers and constant multiples of the identity, the natural suggestion for the correspondence between matrices and complex numbers is the set of all matrices of the form , where .
Rather than using the representation , we can rewrite this as follows:
We will refer to this matrix as the matrix representation of the complex number .
Matrix Representation of Complex Numbers
We can represent the complex number as the matrix
In our first example, we will explore this correspondence in relation to addition of complex numbers.
Example 1: Addition of Complex Numbers Represented as Matrices
Suppose we take the matrix to represent the complex number and the matrix to represent the complex number .
- What is the sum ?
- What does this represent in terms of the complex numbers and ?
Answer
Part 1
We can use the definition of matrix addition to find that
Part 2
Since, , we can see that is the matrix representation of .
Example 2: Multiplication of Complex Numbers Represented as Matrices
Suppose we take the matrix to represent the complex number and the matrix to represent the complex number .
- What is the product ?
- What does it represent?
Answer
Part 1
We can use the definition of matrix multiplication to find that
Part 2
We first consider the product of the two complex numbers
We, therefore, see that is the matrix representation of the product .
The previous example appears to indicate that the correspondence between complex numbers and these types of matrices includes multiplication. However, since we know in general that matrix multiplication is noncommutative, we should check that multiplication of matrices of this type is commutative.
We let be the matrix representation of the complex number and the representation of .
We can now consider the product of the matrix representations as follows:
Similarly, we consider the product :
Hence, we can see that multiplication of matrices of this form is commutative. Furthermore, we can see that multiplying matrix representations of complex numbers directly corresponds to multiplying the complex numbers themselves.
At this point, we might wonder how other operations on complex numbers such as conjugation can be understood in terms of their matrix representation. Alternatively, we might ask the question the other way round: what does the determinant of a matrix represent in terms of complex numbers? In the next example, we will explore this question.
Example 3: Interpreting the Complex Conjugation in terms of Matrices
Let be the matrix representation of the complex number .
- Express the matrix representation of in terms of .
- What does correspond to in relation to ?
- What does correspond to in terms of ?
Answer
Part 1
The matrix representation of is given by The complex conjugate of is given by . We can represent this as a matrix:
This represents the transpose of matrix . Hence, the matrix representation of is .
Part 2
Recall that the determinant of a 2-by-2 matrix is defined as . Therefore,
This corresponds to the sum of the squares of the real and imaginary parts of . Recall that the modulus of is defined as
Hence, corresponds to .
Part 3
We begin by calculating the inverse of . Recall that the inverse of a two-by-two matrix is given by
Therefore,
This corresponds to the complex number which is . Hence, corresponds to the complex number .
Example 4: Interpreting the Matrix Inverse in terms of Complex Numbers
Let be the matrix representation of the complex number . What is the corresponding complex number identity for the matrix identity where is the cofactor matrix of ?
Answer
We begin by recalling some of the correspondences between complex numbers and their matrix representation:
- The determinant of the matrix representation of a complex number corresponds to the square of its modulus.
- The transpose of the matrix representation of a complex number corresponds to complex conjugation.
- The inverse of the matrix representation of a complex number corresponds to the reciprocal of the complex number.
These facts, however, are not sufficient to rewrite the expression as a complex number identity. We need to consider what the cofactor matrix corresponds to. Recall that the cofactor matrix of the two-by-two matrix is given by
Since represents , we have that . Therefore, its cofactor matrix is given by
Hence, the cofactor matrix of is the same as ; that is, . We can now use all these results to give the corresponding complex number expression for
Since matrix inversion corresponds to reciprocation of the complex number, the left-hand side corresponds to . As for the right-hand side, the determinant corresponds to the square of the modulus of , , and since the cofactor matrix is the same as , this corresponds to and its transpose corresponds to conjugation. Therefore, the corresponding complex number identity for is
Now we know that the reciprocal of a complex number corresponds to the inverse of its matrix representation, we can see that complex division can be represented as multiplying the matrix representation of the numerator by the inverse matrix of the denominator.
Example 5: Division of Complex Numbers Represented as Matrices
Let be the matrix representation of the complex number and be the matrix representation of . Determine the matrix representation of .
Answer
There are two ways we could calculate this. We can either use the matrix representations stating that the equivalent matrix expression for is , or we can convert the matrices to their equivalent complex numbers and then undertake the complex division and convert our answer back into its matrix representation at the end. We will demonstrate the first of these methods and then use the second to check our answer.
We begin by calculating the inverse of matrix . Recall that the inverse of a two-by-two matrix is given by
Therefore,
We now multiply this by . Recall that, we do not need to worry about whether we are doing left or right multiplication since the matrices we are dealing with commute. Therefore,
We will check our answer using complex division. The complex number which the matrix represents is ; similarly, is representing . We can now calculate the quotient
Multiplying the numerator and denominator by the complex conjugate of the denominator gives
This is represented by the matrix as expected.
An alternative way to derive the correspondence between complex numbers and matrices is by considering transformations.
There is a direct correspondence between linear maps and matrices: all linear maps can be represented as matrices and all matrices can be interpreted as representing some linear map. In a similar way, we can view multiplication of complex numbers as a linear transformation of the complex plane. To demonstrate the correspondence, we will consider the set of matrices representing rotations about the origin. Let us consider an anticlockwise rotation about the origin through an angle of .
We consider the effect of the transformation on the two basis vectors and . As we can see in the diagram, is mapped to , whereas is mapped to . We can therefore express the matrix of this transformation as
We can rewrite this as
This has striking similarities with Euler’s formula:
In particular, we know that multiplication by represents a rotation through an angle of about the origin as does the matrix . Hence, we can consider to be the matrix representing this transformation. We can further generalize this to include multiplication by a general complex number in the form to find a matrix representation which we can write as which is equivalent to the matrix representation we have already considered:
We can, therefore, see that all matrices of the form represent transformations which consist of a rotation by some angle and a dilation by a factor of .
Example 6: Translating Complex Number Expressions to Matrix Expressions
Let the complex numbers , , and be represented by the matrices , , and . Rewrite the following complex calculation as a matrix calculation:
Answer
Taking complex conjugates corresponds to transposing the matrix representation. Addition of complex numbers corresponds to addition of their matrix representations. Since multiplication of complex numbers corresponds to multiplication of their matrix representations, raising to a power will correspond to raising to an equivalent power. Finally, division corresponds to multiplication by the inverse matrix. Therefore, the complex calculation can be represented as a matrix calculation as follows:
Key Points
- We can represent the complex number as the matrix
- Adding and multiplying complex numbers corresponds to adding or multiplying their matrix representations.
- The determinant of the matrix representation of a complex number corresponds to the square of its modulus.
- The transpose of the matrix representation of a complex number corresponds to complex conjugation.
- The inverse of the matrix representation of a complex number corresponds to the reciprocal of the complex number.
- Dividing by a complex number corresponds to multiplying by the inverse of its matrix representation.
