The equivalence of SVD and eigendecomposition

PCA originally formulated to use eigendecomposition, but we can use SVD instead.

a = rand(4, 3);
a = a - mean(a);
[U, S, V_svd] = svd(a);
[V_eig, D, W ]= eig(a * a');
 
% a == U * S * V_svd'
% a * a' == U * S * V_svd' * V_svd * S' * U'
% V_eig == W
% S.^2 == D (subject to the different ordering of eigenvalues)