%Script file:  See how the SV's change the mapping A
V=eye(2);
U=[cos(pi/6) -sin(pi/6); sin(pi/6) cos(pi/6)];
t=linspace(0, 2*pi);
X=[cos(t);sin(t)];

figure(1)

%S.V.s are 1,1
S=diag([1,1]);
A=U*S*V';
Y=A*X;
subplot(2,2,1)
plot(X(1,:),X(2,:),'k-',Y(1,:),Y(2,:),'r.');
axis equal

%S.V.'s are 3, 1/2
S=diag([3,0.5]);
A=U*S*V';
Y=A*X;
subplot(2,2,2)
plot(X(1,:),X(2,:),'k-',Y(1,:),Y(2,:),'r.');
axis equal

%S.V.'s are 1,0
S=diag([1,0]);
A=U*S*V';
Y=A*X;
subplot(2,2,3)
plot(X(1,:),X(2,:),'k-',Y(1,:),Y(2,:),'r.');
axis equal

%S.V.s are 2, 3
S=diag([2,3]);
A=U*S*V';
Y=A*X;
subplot(2,2,4)
plot(X(1,:),X(2,:),'k-',Y(1,:),Y(2,:),'r.');
axis equal