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Matrix Complex - Matrix Example

This slide is to show the effect of multiplying a complex number with a complex exponential (e^w i). As shown below, mulplication of a complex number (each complex number represented by a colored dots) with a complex exponental result in the rotation of the complex number represented in the cartesian coordinate.

This operation becomes a fundamental building blocks in many of famous equations in engineering. So I strongly recommend you to build up intuitions on this operation.

Followings are the code that I wrote in Octave to creates all the plots shown in this page. You may copy these code and play with these codes. Change variables and try yourself until you get your own intuitive understanding.

< Code 1 >

x = [ ...     

     -1 1; ...

     1  1; ...

     1  -1; ...

     -1 -1; ...

     ];

x = x';

c1 = x(1,:) + j*x(2,:);

c2 = circshift(c1,4);

tstep = pi/20;

t=0:tstep:2*pi;

k=1;

d=0;

e3 = exp(k*t*j)*exp(j*d);

n = 40;

m =  exp(-j*n*2*pi/40);     

tc = m * [c1; c2];

hFig = figure(1,'Position',[300 300 900 250]);    

subplot(1,3,1);

hold on;

plot(real(e3),imag(e3),'k-');

hold on;

plot(real(e3),imag(e3),'bo','MarkerFaceColor',[0 0 1]);

axis([-1.1 1.1 -1.1 1.1]);

xlabel('real : e^{j t}');

ylabel('imaginary : e^{j t}');

plot(real(m),imag(m),'ro',"markerfacecolor",[1 0 0],'MarkerSize',12);

daspect([1 1]);

quiver([0],[0],[real(m)],[imag(m)],"color","red","maxheadsize", 0.1);

line([-1.2 1.2],[0 0],'color','black');

line([0 0],[-1.2 1.2],'color','black');

hold off;

box off;

axis off;

daspect([1 1]);

subplot(1,3,2);

hold on;

plot(real(c2(1)),imag(c2(1)),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',8);

plot(real(c2(2)),imag(c2(2)),'go','MarkerFaceColor',[0 1 0],'MarkerSize',8);

plot(real(c2(3)),imag(c2(3)),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',8);

plot(real(c2(4)),imag(c2(4)),'ko','MarkerFaceColor',[0 0 0],'MarkerSize',8);

axis([-2 2 -2 2]);

daspect([1 1]);

hold off;

grid on;

box on;

subplot(1,3,3);     

hold on;

tc1 = tc(1,:);

plot(real(tc1(1)),imag(tc1(1)),'ro','MarkerFaceColor',[1 0 0],'MarkerSize',8);

plot(real(tc1(2)),imag(tc1(2)),'go','MarkerFaceColor',[0 1 0],'MarkerSize',8);

plot(real(tc1(3)),imag(tc1(3)),'bo','MarkerFaceColor',[0 0 1],'MarkerSize',8);

plot(real(tc1(4)),imag(tc1(4)),'ko','MarkerFaceColor',[0 0 0],'MarkerSize',8);

axis([-2 2 -2 2]);

daspect([1 1]);

hold off;

grid on;

box on;