Complex numbers as matrices

Summary

This lesson associates with each complex number $z = a + bi$ the matrix of the transformation of the plane that multiplies by $z$ , and proves that this correspondence is injective and turns sums into sums and products into products: the matrices of that form make up a faithful copy of the complex numbers inside the square matrices of order two. In that copy, the imaginary unit is the quarter-turn rotation, whose square is the opposite of the identity; the square of the modulus is the determinant, the conjugate is the transpose, and the inverse is the inverse matrix. The polar form decomposes each of those matrices into a rotation followed by a dilation, and Euler’s and De Moivre’s formulas reappear as the rule for composing rotations. The lesson explains why these matrices commute, even though matrices in general do not, and ends with a first look at Hamilton’s quaternions, whose product is no longer commutative.

Learning objectives

Obtain the matrix of multiplication by a complex number and prove that the correspondence between complex numbers and matrices preserves the sum and the product.
Translate into matrices the modulus, the conjugate, the inverse and the polar form of a complex number, and interpret each translation geometrically.
Compute powers, inverses and roots of complex numbers by means of their matrices, and recognize in Euler’s and De Moivre’s formulas the composition of rotations.
Explain why the matrices of complex numbers commute with one another, prove that they are the ones that commute with the quarter-turn rotation, and verify with their matrices that the product of quaternions is not commutative.

The matrix of multiplication by a complex number

In the lesson of the course Algebra and Geometry I on the complex numbers, each complex number $x + yi$ was identified with the point $(x, y)$ of the plane, and it was proved that multiplying by $i$ rotates each point a quarter turn about the origin. If this is so, multiplying by a fixed complex number $z$ is a transformation of the plane, which assigns to each point $w$ the point $zw$ . In the lesson on matrices as transformations of the plane it was proved that a linear transformation is determined by a matrix, whose columns are the images of the vectors $(1, 0)$ and $(0, 1)$ . From this follows the question that guides this lesson: what matrix does multiplication by $z$ have, and what is gained by representing each complex number by that matrix?

PropositionThe matrix of multiplication by z

Let $z = a + bi$ be a fixed complex number. The transformation of the plane that multiplies each complex number $w = x + yi$ by $z$ is linear; its matrix has columns $z$ and $iz$ , and, if $z \neq 0$ , it takes the unit square to a square of side $\abs{z}$ .

We expand the product and read the matrix off it. The figure uses $z = 2 + i$ .

Proof

$zw = (a + bi)(x + yi) = ax + ayi + bxi + b \resaltar{i^2} y = \dato{p}{\resaltar{(ax - by) + (bx + ay) i}}$
We expand the product by distributivity and replace $i^2$ with $-1$ .
$\begin{pmatrix} \resaltar{ax - by} \\ \resaltar{bx + ay} \end{pmatrix} = \dato{m}{\begin{pmatrix} \resaltar{a} & \resaltar{-b} \\ \resaltar{b} & \resaltar{a} \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix}}$
We write the points as column vectors. The coordinates of $zw$ are combinations of $x$ and $y$ with fixed coefficients: this is the product of a matrix by the vector of $w$ , and the transformation is therefore linear.
$z \cdot \resaltar{1} = \resaltar{a + bi}, \qquad z \cdot \resaltar{i} = \resaltar{-b + ai}$
The columns of the matrix are the images of $1$ and of $i$ . The first is $z$ ; the second is $iz = -b + ai$ , the rotation of $z$ by a quarter turn.
$\abs{iz} = \abs{i} \abs{z} = \resaltar{\abs{z}}, \qquad \resaltar{z \cdot (1 + i)} = \resaltar{z + iz}$
The unit square, with vertices $0$ , $1$ , $1 + i$ and $i$ , is transformed into the quadrilateral with vertices $0$ , $z$ , $z + iz$ and $iz$ . Its sides from the origin have the same length $\abs{z}$ and are perpendicular: the image is a square of side $\abs{z}$ , rotated with respect to the unit square.

The matrix obtained has a very special form: its two diagonal entries are equal, and the other two are opposite. The matrix of the complex number $z = a + bi$ , with $a$ and $b$ real, is the matrix of multiplication by $z$ :

M(z) = \begin{pmatrix} a & -b \\ b & a \end{pmatrix}

The first column of $M(z)$ is $z$ itself, and the second is $iz$ . On separating the real and imaginary parts of the complex number, the matrix decomposes into a combination of two fixed matrices: the identity $I$ , which is the matrix of the number $1$ , and the matrix $J$ of the number $i$ .

M(a + bi) = a \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} + b \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = aI + bJ

Specifically, the real numbers correspond to the multiples of the identity, $M(a) = aI$ , which are the dilations with center at the origin, and the purely imaginary numbers $bi$ to the multiples of $J$ . The first entries of the table of correspondences are these.

M(1) = I, \qquad M(i) = J, \qquad M(-1) = -I, \qquad M(-i) = -J, \qquad M(0) = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}

In the rest of this lesson

Sum and product are preserved
The imaginary unit as a quarter turn
Determinant, transpose and inverse
The polar form: a rotation and a dilation
Euler and De Moivre as composition of rotations
Why these matrices commute
A look at the quaternions
Solved problems
A linear system disguised as a division
The eighth power of 1 + i, by rotating
The matrices that commute with the imaginary unit
Infinitely many square roots of minus the identity
The square roots of a rotation
Proposed problems
Rotations are the matrices of modulus one
The quadratic equation of the matrix of a complex number
The path of a graphics turtle
The hyperbolic numbers
Two quaternions that do not commute