%%
clear 
rng(1)
format

%%
n = 4;
m = 8;
A = rand(n, m)

%%
[U, S, V] = svd(A)
norm(U*S*V' - A)

%%
s = S(:,1:n)
v = V(:,1:n)

%%
norm(U*s*v' - A)

%%
norm(U(:,1)*s(1,1)*v(:,1)' + U(:,2)*s(2,2)*v(:,2)' + U(:,3)*s(3,3)*v(:,3)' + U(:,4)*s(4,4)*v(:,4)'- A)