Using this code in Matlab:

^{clear all; close all; % clear workspace}

^{%% Dataset is created}

^{N=200; % number of data points}

^{x1=10*randn(N,1); % first coordinates}

^{x2=4+x1+10*randn(N,1); % second coordinates - example in space of dimension D=2}

^{%% pca}

^{X=[(x1-mean(x1))';(x2-mean(x2))'];}

^S=(X*X'/N);

^{[U,L,V] = svd(S);}

^{%% plot data}

^{figure;plot(x1,x2,'kx','LineWidth',2,'MarkerFaceColor','k'),...,}

^{hold on, plot(mean(x1),mean(x2),'ro','LineWidth',2,'MarkerFaceColor','r'), hold on,...}

^{quiver(mean(x1),mean(x2),V(1,1),V(2,1),sqrt(L(1,1)),'g','LineWidth',2), hold on, ...}

^{quiver(mean(x1),@mean(x2),V(1,2),V(2,2),sqrt(L(2,2)),'b','LineWidth',2),hold off, axis equal}

Graph:

Consider u1 and u2 eigenvectors of matrix S wih respective eigenvalues 1 and 2 with 1 > 2. Defining u3 = u1 + u2, is u3 an eigenvector of S? Explain.