% the script VisualizeSVD_LE shows the effects of a 3X2 random
% matrix acting on the unit circle and visualizes the SVD
clf
% unit vectors on the unit circle (in 2D)
theta=linspace(0,2*pi,100);x=cos(theta); y=sin(theta); 
% random matrix A 
A=rand(3,2) 
% columns of B contains the coordinate (in 3D) of the vectors
% obtained after the transformation operated by A
B=A*[x;y];
[U,S,V]=svd(A);
% scaling of the columns of U by means of singular values (be careful, s3=0)
UU=U*S;
% visualize vectors on the unit circle
figure; plot(x,y,'.b'); ylim([-1.1,1.1]);axis equal;  hold on; 
% visualize column vectors of V 
% (are an orthogonal basis in R2)
disp('visualize vectors on the unit circle')
disp('column vectors of V are an orthogonal basis in R2')
title('column vectors of V, basis in R2',FontSize=9)
quiver(zeros(1,2),zeros(1,2), V(1,:),V(2,:));
set(gcf, 'Position', [100, 100, 400, 300]); % modify the size of the figure
axis equal; hold off
% visualize vectors after the transformation by A. They form an ellipse in
% R3.
figure; plot3(B(1,:),B(2,:),B(3,:),'.r'); axis equal; hold on
% The ellipse lies on the plane generated by the first two columns of U
% (that are an orthogonal basis on such plane representing the range of A.
% The length of the largest semiaxis is s1; the length of the minor
% semiaxis is s2.
quiver3(0,0,0,UU(1,1),UU(2,1),UU(3,1))
quiver3(0,0,0,UU(1,2),UU(2,2),UU(3,2))
quiver3(0,0,0,U(1,3),U(2,3),U(3,3),'g')
axis equal; hold off
disp('the ellipse is the transformation of the unit circle performed by A')
disp('the ellipse lies on the plane spanned by the first two columns of U')
disp('column of U = orthogonal basis on the (plane) range of A')
disp('the length of major semiaxis is sigma(1), the length of minor semiaxis is sigma(2)')
disp('the third column of U is the basis of the orthogonal complement of the range of A')