Plot Complex Numbers- MATLAB & Simulink (2024)

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This example shows how to plot complex numbers in MATLAB®. A complex number $z$ is a number that can be written in the form

$z = x + y i$ ,

where $x$ and $y$ are real numbers, and $i$ is the imaginary unit, which is defined as $i^{2} = - 1$ . The number $x$ is the real part of the complex number, which is denoted by $x = R e (z)$ , and the number $y$ is the imaginary part of the complex number, which is denoted by $y = I m (z)$ . You can plot a complex number as a pair of coordinates $(x, y)$ on the complex plane, also known as the Argand diagram. This diagram uses the Cartesian coordinates to represent the real part in the $x$ -axis and the imaginary part in the $y$ -axis.

You can also represent a complex number using the polar representation. The complex number is written in the form

$z = r e^{i θ} = r (\cos θ + i \sin θ)$ ,

where $r$ is the absolute value or magnitude of the complex number, and $θ$ is the phase angle of the complex number. In this representation, you can plot a complex number as a point in the polar coordinates with radius $r$ (the distance from the origin) and polar angle $θ$ (the counterclockwise angle between the positive real axis and the line connecting the point to the origin).

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 equalgrid onxlabel("Re(z)")ylabel("Im(z)")

Plot Complex Roots of Unity in Cartesian Coordinates

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

$z^{n} = 1$ ,

where $n$ is a positive integer.

The $n$ th roots of unity are

$\exp (\frac{2 k π i}{n}) = \cos \frac{2 k π}{n} + i \sin \frac{2 k π}{n}$ , for $k = 0, 1, \dots, n - 1$ .

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 $p_{1} x^{n} + \dots + p_{n} x + p_{n + 1} = 0$ . 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 equalgrid onxlabel("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

$z = f (t) = t \exp (i t)$

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

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 equalgrid onxlabel("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 equalgrid onxlabel("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 equalgrid onlegend("z1","z2")xlabel("Re(z)")ylabel("Im(z)")

FAQs

Can you plot complex numbers in MATLAB? ›

You can plot a complex number as a pair of coordinates ( x , y ) on the complex plane, also known as the Argand diagram.

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How to solve complex numbers in MATLAB? ›

You can use i to enter complex numbers. You also can use the character j as the imaginary unit. To create a complex number without using i and j , use the complex function. z = a + b i returns a complex numerical constant, z .

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How to plot a complex number? ›

Given a complex number, represent its components on the complex plane.

Determine the real part and the imaginary part of the complex number.
Move along the horizontal axis to show the real part of the number.
Move parallel to the vertical axis to show the imaginary part of the number.
Plot the point.

Jun 25, 2022

Keep Reading ›

How to get imaginary part of complex number in MATLAB? ›

Y = imag( Z ) returns the imaginary part of each element in array Z .

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What is the representation of complex numbers in MATLAB? ›

Complex numbers, whose set is denoted by ℂ, have the form z=x+iy, or z=x+jy, where i, also denoted as j, is the imaginary unit vector on the complex plane ℂ, in a positive vertical direction; so j² = -1. matlab utelizes both standard notations for such unit vector on complex plane: 1i and 1j .

What is an example of a complex function in MATLAB? ›

Assign a complex constant to the variable or use the complex function. For example: x = 5 + 6i; % x is a complex number by assignment. y = complex(5,6); % y is the complex number 5 + 6i.

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How do you simplify complex numbers in MATLAB? ›

In most cases, to simplify a symbolic expression using Symbolic Math Toolbox™, you only need to use the simplify function. But for some large and complex expressions, you can obtain a faster and simpler result by using the expand function before applying simplify .

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How to generate random complex numbers in MATLAB? ›

X = randn(___,"like", p ) returns an array of random numbers like p ; that is, of the same data type and complexity (real or complex) as p . You can specify either typename or "like" , but not both. X = randn( s ,___) generates numbers from random number stream s instead of the default global stream.

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How do you make a complex plot? ›

Here are seven simple and easy practices that can help you tackle seemingly complicated plots and stories.

Don't Map Everything Out. ...
Surprise Yourself. ...
Know Your Script's Bad Trailer Moments. ...
Don't Lie, Raise Questions. ...
Don't Oversell the Twists, Turns, and Reveals. ...
Don't Overexplain Everything. ...
Pepper Your Script.

Nov 28, 2022

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Can you graph complex functions? ›

Graphing a complex function is difficult because you need 2 (real) dimensions for the domain and 2 (real) dimensions for the range - a total of 4 dimensions. In the Complex Function Grapher app, the domain of a complex function is graphed on the base plane. The range is graphed using polar coordinates.

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How to initialize complex numbers in MATLAB? ›

z = complex( a , b ) creates a complex output, z , from two real inputs, such that z = a + bi .

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What is the nth root of a complex number in MATLAB? ›

This is just a MATLAB function that finds the nth root of a complex number in x +iy form. Essentially, (x+iy)^1/n. It then plots the roots in a polar plane.

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How do you declare a complex in MATLAB? ›

Declaring Complex Signals

x and y are declared by complex constant assignment; z is created using the using the complex() function. function [x,y,z] = fcn % create 8 bit complex constants x = uint8(1 + 2i); y = uint8(3 + 4j); z = uint8(complex(5, 6));

Plot Complex Numbers - MATLAB & Simulink (2024)