%
% Problem epage 70
% Example epage 51
% 
% Written by:
% -- 
% John L. Weatherwax                2005-08-14
% 
% email: wax@alum.mit.edu
% 
% Please send comments and especially bug reports to the
% above email address.
% 
%-----

close all; drawnow; 
clc; 
clear; 

rand('seed',0); randn('seed',0); 

n = 10; 
p = 5; 
X = rand(n,p);

m1 = (X.') * X; 
m2 = X * (X.');

[eigvec1,eigval1] = eig( m1 ); eigval1 = sort(diag(eigval1)); 
[eigvec2,eigval2] = eig( m2 ); eigval2 = sort(diag(eigval2)); 

fprintf('displaying the non-zero eigenvalues...\n');
flipud([ eigval1(:), eigval2(end-4:end) ]) % <- note this assumes that the eigenvalues are sorted in increasing/ascending order

[u,s,v] = svd( X );                        % <- assuming vectors are returned in decending order
% get the diagonal elemenst of s:
s_diag = diag(s); 

fprintf( 'the square of these sigular values are given by \n' ); 
(s_diag.^2).'

fprintf( 'the eigenvectors of X^T X (permuted) and the right singular vectors V are\n' ); 
disp(eigvec1(:,p:-1:1))
disp(-v) 
fprintf( 'they are the same appart from sign\n\n' ); 

% flip the ordering of the eigenvectors so that they correspond to a decreasing eigenvalue ordering: 
eigvec1 = fliplr(eigvec1); eigvec2 = fliplr(eigvec2); 

fprintf( 'the first p=%d eigenvectors of X X^T (permuted) and the left singular vectors U are\n', p ); 
disp(eigvec2(:,1:p))
disp(u(:,1:p)) 
fprintf( 'they are the same appart from sign\n\n' );