Explainer: Matrix Representation of Complex Numbers

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 𝐴 and 𝐡, 𝐴𝐡≠𝐡𝐴 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 𝐼 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, 𝐼𝐴=𝐴𝐼=𝐴,𝐴𝐴=𝐴𝐴=𝐼, and 𝐼=𝐼. In fact, there is a direct correspondence between the set of all matrices of the form π‘πΌοŠ for a constant π‘βˆˆβ„ 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 𝑀=0βˆ’110.

Using matrix multiplication, we have 𝑀=0βˆ’1100βˆ’110=(0)(0)+(βˆ’1)(1)(0)(βˆ’1)+(βˆ’1)(0)(1)(0)+(0)(1)(1)(βˆ’1)+(0)(0)=ο”βˆ’100βˆ’1.

Interestingly, this is equal to the negative identity matrix. Hence, 𝑀=βˆ’πΌοŠ¨οŠ¨. 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 π‘Ž such that π‘Ž=βˆ’1. However, there is a complex number 𝑖 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 𝑀 to correspond to 𝑖, it makes sense that the imaginary numbers would correspond to constant multiples of 𝑀. 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 π‘ŽπΌ+π‘π‘€οŠ¨, where π‘Ž,π‘βˆˆβ„.

Rather than using the representation π‘ŽπΌ+π‘π‘€οŠ¨, we can rewrite this as follows: π‘ŽπΌ+𝑏𝑀=π‘Žο”1001+𝑏0βˆ’110=ο”π‘Ž00π‘Žο +0βˆ’π‘π‘0=ο”π‘Žβˆ’π‘π‘π‘Žο .

We will refer to this matrix as the matrix representation of the complex number π‘Ž+𝑏𝑖.

Matrix Representation of Complex Numbers

We can represent the complex number π‘Ž+𝑏𝑖 as the matrix ο”π‘Žβˆ’π‘π‘π‘Žο .

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 𝑀=5βˆ’775 to represent the complex number 𝑧=5+7π‘–οŠ§ and the matrix 𝑁=18βˆ’81 to represent the complex number 𝑧=1βˆ’8π‘–οŠ¨.

  1. What is the sum 𝑀+𝑁?
  2. What does this represent in terms of the complex numbers π‘§οŠ§ and π‘§οŠ¨?

Answer

Part 1

We can use the definition of matrix addition to find that 𝑀+𝑁=5βˆ’775+18βˆ’81=5+1βˆ’7+87βˆ’85+1=61βˆ’16.

Part 2

Since, 𝑧+𝑧=6βˆ’π‘–οŠ§οŠ¨, we can see that 𝑀+𝑁 is the matrix representation of 𝑧+π‘§οŠ§οŠ¨.

Example 2: Multiplication of Complex Numbers Represented as Matrices

Suppose we take the matrix 𝑁=34βˆ’43 to represent the complex number 3βˆ’4𝑖 and the matrix 𝑀=2βˆ’552 to represent the complex number 2+5𝑖.

  1. What is the product 𝑀𝑁?
  2. What does it represent?

Answer

Part 1

We can use the definition of matrix multiplication to find that 𝑀𝑁=2βˆ’55234βˆ’43=(2)(3)+(βˆ’5)(βˆ’4)(2)(4)+(βˆ’5)(3)(5)(3)+(2)(βˆ’4)(5)(4)+(2)(3)=26βˆ’7726.

Part 2

We first consider the product of the two complex numbers (3βˆ’4𝑖)(2+5𝑖)=6+15π‘–βˆ’8π‘–βˆ’20𝑖=26+7𝑖.

We, therefore, see that 𝑀𝑁 is the matrix representation of the product (3βˆ’4𝑖)(2+5𝑖).

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 π‘€οŠ§ be the matrix representation of the complex number 𝑧=π‘Ž+π‘π‘–οŠ§ and π‘€οŠ¨ the representation of 𝑧=𝑐+π‘‘π‘–οŠ¨.

We can now consider the product π‘€π‘€οŠ§οŠ¨ of the matrix representations as follows: 𝑀𝑀=ο”π‘Žβˆ’π‘π‘π‘Žο ο”π‘βˆ’π‘‘π‘‘π‘ο =ο”π‘Žπ‘βˆ’π‘π‘‘βˆ’π‘Žπ‘‘βˆ’π‘π‘π‘Žπ‘‘+π‘π‘π‘Žπ‘βˆ’π‘π‘‘ο .

Similarly, we consider the product π‘€π‘€οŠ¨οŠ§: ο”π‘βˆ’π‘‘π‘‘π‘ο ο”π‘Žβˆ’π‘π‘π‘Žο =ο”π‘Žπ‘βˆ’π‘π‘‘βˆ’π‘Žπ‘‘βˆ’π‘π‘π‘Žπ‘‘+π‘π‘π‘Žπ‘βˆ’π‘π‘‘ο .

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 𝑀 be the matrix representation of the complex number 𝑧=π‘Ž+𝑏𝑖.

  1. Express the matrix representation of π‘§βˆ— in terms of 𝑀.
  2. What does det𝑀 correspond to in relation to 𝑧?
  3. What does π‘€οŠ±οŠ§ correspond to in terms of 𝑧?

Answer

Part 1

The matrix representation of 𝑧=π‘Ž+𝑏𝑖 is given by 𝑀=ο”π‘Žβˆ’π‘π‘π‘Žο . The complex conjugate of 𝑧 is given by 𝑧=π‘Žβˆ’π‘π‘–βˆ—. We can represent this as a matrix: ο”π‘Žπ‘βˆ’π‘π‘Žο .

This represents the transpose of matrix 𝑀. Hence, the matrix representation of π‘§βˆ— is 𝑀T.

Part 2

Recall that the determinant of a 2-by-2 matrix 𝐴=ο”π‘Žπ‘π‘π‘‘ο  is defined as det𝐴=π‘Žπ‘‘βˆ’π‘π‘. Therefore, det𝑀=π‘Žβˆ’ο€Ήβˆ’π‘ο…=π‘Ž+𝑏.

This corresponds to the sum of the squares of the real and imaginary parts of 𝑧. Recall that the modulus of 𝑧=π‘Ž+𝑏𝑖 is defined as |𝑧|=βˆšπ‘Ž+𝑏.

Hence, det𝑀 corresponds to |𝑧|.

Part 3

We begin by calculating the inverse of 𝑀. Recall that the inverse of a two-by-two matrix 𝐴=ο”π‘Žπ‘π‘π‘‘ο  is given by 𝐴=1π‘Žπ‘‘βˆ’π‘π‘ο“π‘‘βˆ’π‘βˆ’π‘π‘ŽοŸ.

Therefore, 𝑀=1π‘Ž+π‘ο”π‘Žπ‘βˆ’π‘π‘Žο .

This corresponds to the complex number π‘Žπ‘Ž+π‘βˆ’π‘π‘Ž+π‘π‘–οŠ¨οŠ¨οŠ¨οŠ¨ which is 1𝑧. Hence, π‘€οŠ±οŠ§ corresponds to the complex number 1𝑧.

Example 4: Interpreting the Matrix Inverse in terms of Complex Numbers

Let 𝑀 be the matrix representation of the complex number 𝑧=π‘Ž+𝑏𝑖. What is the corresponding complex number identity for the matrix identity 𝑀=1𝑀𝐢,detT where 𝐢 is the cofactor matrix of 𝑀?

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 𝐴=ο”π‘Žπ‘π‘π‘‘ο  is given by 𝐢=ο”π‘‘βˆ’π‘βˆ’π‘π‘Žο .

Since 𝑀 represents 𝑧=π‘Ž+𝑏𝑖, we have that 𝑀=ο”π‘Žβˆ’π‘π‘π‘Žο . Therefore, its cofactor matrix is given by 𝐢=ο”π‘Žβˆ’π‘π‘π‘Žο .

Hence, the cofactor matrix of 𝑀 is the same as 𝑀; that is, 𝐢=𝑀. We can now use all these results to give the corresponding complex number expression for 𝑀=1𝑀𝐢.detT

Since matrix inversion corresponds to reciprocation of the complex number, the left-hand side corresponds to 1𝑧. As for the right-hand side, the determinant corresponds to the square of the modulus of 𝑧, |𝑧|, and since the cofactor matrix is the same as 𝑀, this corresponds to 𝑧 and its transpose corresponds to conjugation. Therefore, the corresponding complex number identity for 𝑀=1π‘€πΆοŠ±οŠ§detT is 1𝑧=𝑧|𝑧|.βˆ—οŠ¨

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 𝑀=711βˆ’117 be the matrix representation of the complex number π‘§οŠ§ and 𝑀=4βˆ’114 be the matrix representation of π‘§οŠ¨. Determine the matrix representation of π‘§π‘§οŠ§οŠ¨.

Answer

There are two ways we could calculate this. We can either use the matrix representations stating that the equivalent matrix expression for π‘§π‘§οŠ§οŠ¨ is π‘€π‘€οŠ§οŠ±οŠ§οŠ¨, 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 π‘€οŠ¨. Recall that the inverse of a two-by-two matrix 𝐴=ο”π‘Žπ‘π‘π‘‘ο  is given by 𝐴=1π‘Žπ‘‘βˆ’π‘π‘=ο“π‘‘βˆ’π‘βˆ’π‘π‘ŽοŸ.

Therefore, 𝑀=14+141βˆ’14=11741βˆ’14.

We now multiply this by π‘€οŠ§. 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, 𝑀𝑀=117711βˆ’11741βˆ’14=1177Γ—4+11Γ—(βˆ’1)7Γ—1+11Γ—4(βˆ’11)Γ—4+7Γ—(βˆ’1)(βˆ’11)Γ—1+4Γ—7=1171751βˆ’5117=13βˆ’31.

We will check our answer using complex division. The complex number π‘§οŠ§ which the matrix 𝑀=711βˆ’117 represents is 𝑧=7βˆ’11π‘–οŠ§; similarly, 𝑀=4βˆ’114 is representing 𝑧=4+π‘–οŠ¨. We can now calculate the quotient 𝑧𝑧=7βˆ’11𝑖4+𝑖.

Multiplying the numerator and denominator by the complex conjugate of the denominator gives 𝑧𝑧=(7βˆ’11𝑖)(4βˆ’π‘–)(4+𝑖)(4βˆ’π‘–)=28βˆ’7π‘–βˆ’44𝑖+11𝑖4+1=17βˆ’51𝑖17=1βˆ’3𝑖.

This is represented by the matrix 13βˆ’31 as expected.

An alternativeway 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 viewmultiplication 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 πœƒ.

We consider the effect of the transformation on the two basis vectors ⟨1,0⟩ and ⟨0,1⟩. As we can see in the diagram, ⟨1,0⟩ is mapped to βŸ¨πœƒ,πœƒβŸ©cossin, whereas ⟨0,1⟩ is mapped to βŸ¨βˆ’πœƒ,πœƒβŸ©sincos. We can therefore express the matrix of this transformation as 𝑅=ο”πœƒβˆ’πœƒπœƒπœƒο .cossinsincos

We can rewrite this as 𝑅=πœƒο”1001+πœƒο”0βˆ’110.cossin

This has striking similarities with Euler’s formula: 𝑒=πœƒ+π‘–πœƒ.cossin

In particular, we know that multiplication by 𝑒 represents a rotation through an angle of πœƒ about the origin as does the matrix 𝑅. Hence, we can consider 𝑅 to be the matrix representing this transformation. We can further generalize this to include multiplication by a general complex number in the form π‘Ÿπ‘’οƒοΌ to find a matrix representation which we can write as ο”π‘Ÿπœƒβˆ’π‘Ÿπœƒπ‘Ÿπœƒπ‘Ÿπœƒο cossinsincos which is equivalent to the matrix representation we have already considered: ο”π‘Žβˆ’π‘π‘π‘Žο .

We can, therefore, see that all matrices of the form ο”π‘Žβˆ’π‘π‘π‘Žο  represent transformations which consist of a rotation by some angle πœƒ and a dilation by a factor of π‘Ÿ.

Example 6: Translating Complex Number Expressions to Matrix Expressions

Let the complex numbers π‘§οŠ§, π‘§οŠ¨, and π‘§οŠ© be represented by the matrices π‘€οŠ§, π‘€οŠ¨, and π‘€οŠ©. Rewrite the following complex calculation as a matrix calculation: (𝑧+𝑧)𝑧.βˆ—οŠ§οŠ¨οŠ¨οŠ©

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 (𝑧+𝑧)π‘§βˆ—οŠ§οŠ¨οŠ¨οŠ© can be represented as a matrix calculation as follows: 𝑀+𝑀𝑀.T

Key Points

  • We can represent the complex number π‘Ž+𝑏𝑖 as the matrix ο”π‘Žβˆ’π‘π‘π‘Žο .
  • 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.

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