1.6: Matrix Representation of Complex Numbers

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In our studies of complex numbers, we noted that multiplication of a complex number by \(e^{iθ}\) rotates that complex number an angle \(θ\) in the complex plane. This leads to the idea that we might be able to represent complex numbers as matrices with \(e^{iθ}\) as the rotation matrix.

Accordingly, we begin by representing \(e^{iθ}\) as the rotation matrix, that is,

\[\begin{aligned}e^{i\theta}&=\left(\begin{array}{rr}\cos\theta&-\sin\theta \\ \sin\theta&\cos\theta\end{array}\right) \\ &=\cos\theta\left(\begin{array}{cc}1&0\\0&1\end{array}\right)+\sin\theta\left(\begin{array}{rr}0&-1\\1&0\end{array}\right).\end{aligned} \nonumber \]

Since \(e^{iθ} = \cos θ + i \sin θ\), we are led to the matrix representations of the unit numbers as

\[1=\left(\begin{array}{cc}1&0\\0&1\end{array}\right),\quad i=\left(\begin{array}{rr}0&-1\\1&0\end{array}\right).\nonumber \]

A general complex number \(z = x + iy\) is then represented as

\[z=\left(\begin{array}{rr}x&-y\\y&x\end{array}\right).\nonumber \]

The complex conjugate operation, where \(i → −i\), is seen to be just the matrix transpose.

Example \(\PageIndex{1}\)

Show that \(i^2=-1\) in the matrix representation.

Solution

We have

\[i^2=\left(\begin{array}{rr}0&-1\\1&0\end{array}\right)\left(\begin{array}{rr}0&-1\\1&0\end{array}\right)=\left(\begin{array}{rr}-1&0\\0&-1\end{array}\right)=-\left(\begin{array}{cc}1&0\\0&1\end{array}\right)=-1.\nonumber \]

Example \(\PageIndex{2}\)

Show that \(z\overline{z}=x^2+y^2\) in the matrix representation.

Solution

We have

\[z\overline{z}=\left(\begin{array}{rr}x&-y\\y&x\end{array}\right)\left(\begin{array}{rr}x&y\\-y&x\end{array}\right)=\left(\begin{array}{cc}x^2+y^2&0\\0&x^2+y^2\end{array}\right)=(x^2+y^2)\left(\begin{array}{cc}1&0\\0&1\end{array}\right)=(x^2+y^2).\nonumber \]

We can now see that there is a one-to-one correspondence between the set of complex numbers and the set of all two-by-two matrices with equal diagonal elements and opposite signed off-diagonal elements. If you do not like the idea of \(\sqrt{-1}\), then just imagine the arithmetic of these two-by-two matrices!