clear
close all

alpha =-1;
beta = 1;
lambda1 = alpha+i*beta;
lambda2 = alpha-i*beta;
Lambda = diag([lambda1;lambda2]);

a = [1;0];
b = [0;2];
v1 = a+i*b;
v2 = a-i*b;
T = [v1,v2];

A = T*Lambda*inv(T);

[Z1,Z2] = meshgrid(-10:1:10,-10:1:10);
Uz = alpha*Z1+beta*Z2;
Vz = - beta*Z1+alpha*Z2;
subplot(1,2,2)
quiver(Z1,Z2,Uz,Vz);
xlabel('z_1')
ylabel('z_2')
axis([-10,10,-10,10])
axis square
hold on

[X1,X2] = meshgrid(-10:1:10,-10:1:10);
Ux = A(1,1)*X1+A(1,2)*X2;
Vx = A(2,1)*X1+A(2,2)*X2;
subplot(1,2,1)
quiver(X1,X2,Ux,Vx);
xlabel('x_1')
ylabel('x_2')
axis([-10,10,-10,10])
axis square
hold on

t = [0:0.01:5]';
for theta = 0:pi/4:2*pi
z0 = 5*[cos(theta);sin(theta)];

z1 = exp(alpha*t).*(cos(beta*t)*z0(1)+sin(beta*t)*z0(2));
z2 = exp(alpha*t).*(-sin(beta*t)*z0(1)+cos(beta*t)*z0(2));

figure(1)
subplot(1,2,2)
plot(z1(1),z2(1),'ro',z1,z2,'r')
hold on

x = z1*a'+z2*b';
x1 = x(:,1);
x2 = x(:,2);

subplot(1,2,1)
plot(x1(1),x2(1),'ro',x1,x2,'r')
hold on

figure(2)
subplot(2,2,2)
plot(t(1),z1(1),'ro',t,z1,'r')
ylabel('z_1(t)')
xlabel('t (s)')
hold on
subplot(2,2,4)
plot(t(1),z2(1),'ro',t,z2,'r')
ylabel('z_2(t)')
xlabel('t (s)')
hold on

subplot(2,2,1)
plot(t(1),x1(1),'ro',t,x1,'r')
ylabel('x_1(t)')
xlabel('t (s)')
hold on
subplot(2,2,3)
plot(t(1),x2(1),'ro',t,x2,'r')
ylabel('x_2(t)')
xlabel('t (s)')
hold on

end