Plot Complex Numbers

Plot Array of Complex Numbers

Create a vector that contains the complex numbers 3 + 4i, -4 - 3i, 1 - 2i, and -1 - 1i.

z = [3 + 4i; -4 - 3i; 1 - 2i; -1 - 1i]

z = 4×1 complex

   3.0000 + 4.0000i
  -4.0000 - 3.0000i
   1.0000 - 2.0000i
  -1.0000 - 1.0000i

Plot the imaginary part against the real part of the complex vector z by using plot. Use the real function and imag function to return the real and imaginary parts of the complex vector, respectively.

plot(real(z),imag(z),"o")
axis equal
grid on
xlabel("Re(z)")
ylabel("Im(z)")

You can also use plot(z,LineSpec) instead of plot(real(z),imag(z),LineSpec) to plot an array of complex numbers. This function automatically plots the real part in the $$x$$-axis and the imaginary part in the $$y$$-axis.

Plot Complex Roots of Unity in Cartesian Coordinates

The $$n$$th roots of unity are complex numbers that satisfy the polynomial equation

${\mathit{z}}^{\mathit{n}}=1$,

where $$n$$ is a positive integer.

The $$n$$th roots of unity are

$\exp \left(\frac{2\mathit{k}\pi \mathrm{i}}{\mathit{n}}\right)=\cos \frac{2\mathit{k}\pi }{\mathit{n}}+\mathrm{i}\sin \frac{2\mathit{k}\pi }{\mathit{n}}$, for $\mathit{k}=0,1,\dots ,\mathit{n}-1$.

To find the complex roots of unity, you can solve the polynomial equation by using roots. The roots function solves polynomial equations of the form ${\mathit{p}}_1{\mathit{x}}^{\mathit{n}}+\cdots +{\mathit{p}}_{\mathit{n}}\mathit{x}+{\mathit{p}}_{\mathit{n}+1}=0$. For example, find the fifth roots of unity of $$z^5=1$$, or $$z^5-1=0$$.

p = [1 0 0 0 0 -1];
z = roots(p)

z = 5×1 complex

  -0.8090 + 0.5878i
  -0.8090 - 0.5878i
   0.3090 + 0.9511i
   0.3090 - 0.9511i
   1.0000 + 0.0000i

Plot the complex roots of unity in the Cartesian coordinates.

plot(z,"o")
axis equal
grid on
xlabel("Re(z)")
ylabel("Im(z)")

Plot Complex Numbers in Polar Coordinates

Plot the fifth roots of unity in the polar coordinates by using polarplot. Use the angle function to return the phase angles of the complex roots, and use the abs function to return the absolute values or radii of the complex roots.

polarplot(angle(z),abs(z),"o")

You can also use polarplot(z,LineSpec) instead of polarplot(angle(z),abs(z),LineSpec) to plot an array of complex numbers in the polar coordinates. This function automatically plots the radii and phase angles of the complex numbers.

Plot Parametric Curve in Complex Plane

Define a parametric curve that has the form

with the parameter $$t$$ in the interval $$[0,4\pi ]$$.

Create a vector t of 200 equally spaced points within this interval to parameterize $$t$$. Define the points that lie on the complex curve as a complex vector z.

t = linspace(0,4*pi,200);
z = t.*exp(1i*t);

Plot the complex curve in the Cartesian coordinates.

plot(z,"-")
axis equal
grid on
xlabel("Re(z)")
ylabel("Im(z)")

Plot the complex curve in the polar coordinates.

polarplot(z,"-")

Plot Eigenvalues of Square Matrix

A real $$n$$-by-$$n$$ square matrix has $$n$$ eigenvalues (counting algebraic multiplicities) that either are real or occur in complex conjugate pairs.

For example, consider a 20-by-20 real matrix with random elements that are sampled from a standard normal distribution. Calculate the eigenvalues using eig.

rng("default")
z = eig(randn(20));

Plot the imaginary part against the real part of all 20 eigenvalues. Notice that for each eigenvalue $$z_k=x_k+y_{k}i$$ that is not on the real axis, there is another complex conjugate pair of this eigenvalue $$z_k^{*}=x_k-y_{k}i$$.

plot(z,"o")
axis equal
grid on
xlabel("Re(z)")
ylabel("Im(z)")

Plot Multiple Complex Data Sets

Plot the imaginary part against the real part of two complex data sets. If you pass multiple complex input arguments to plot, such as plot(z1,z2), then the plot function ignores the imaginary part and plots only the real part of the inputs. To plot the real part against the imaginary part for multiple complex inputs, you must explicitly pass the real part and the imaginary part to plot.

For example, create two complex vectors z1 and z2.

x = -2:0.25:2;
z1 = x.^exp(-x.^2);
z2 = 2*x.^exp(-x.^2);

Find the real part and imaginary part of each vector by using the real and imag functions.

re_z1 = real(z1);
im_z1 = imag(z1);
re_z2 = real(z2);
im_z2 = imag(z2);

Plot the complex data.

plot(re_z1,im_z1,"*",re_z2,im_z2,"o")
axis equal
grid on
legend("z1","z2")
xlabel("Re(z)")
ylabel("Im(z)")