SVD_demo_with_cov_matrix.m

%% The SVD demo with Covariance matrix


clear all;
clc;

% origin_mat = [2 4 6 8; 9 8 0 1];                %First original matrix 2x4
origin_mat = [2 4 6 8; 9 8 0 1; 5 3 -2 -3];       %Second original matrix 3x4

[u1, s1, v1]=svd(origin_mat);

sv=s1*v1';

origin_mat_hat=u1*sv;                              % reconstructed matrix


%% Now SVD with covariance matrix
  
%c = cov(origin_mat,origin_mat);     % Try with first original matrix 2x4
c = origin_mat*origin_mat';          % Try with second original matrix 3x4

[u2,s2,v2]=svd(c);

p=u2'*origin_mat;

origin_mat_hat_cov = u2*p;


%% Reversing sign

%c = cov(origin_mat,origin_mat);    
c = origin_mat*origin_mat';                % This form is easy to work with

[u2,s2,v2]=svd(c);

u2_rev_sign = u2 * -1;                     % reversing sign

p=u2_rev_sign'*origin_mat;


origin_mat_hat_cov_rev = u2_rev_sign * p;

%% Removing negativity

%c = cov(origin_mat,origin_mat);    
c = origin_mat*origin_mat';            

[u2,s2,v2]=svd(c);

u2_unsignd = abs(u2);                      % removing all negative sign

p=u2_unsignd'*origin_mat;


origin_mat_hat_cov_unsigned = u2_unsignd*p;

%% Summary
disp('------------------Summary--------------------------')
origin_mat
origin_mat_hat
origin_mat_hat_cov
origin_mat_hat_cov_rev
origin_mat_hat_cov_unsigned