Projection of a vector $v$ onto the column space of a matrix $A$ is given by $AA^\dagger v$. From the definition of Moore-Penrose Inverse we know that $AA^\dagger v = (A^T)^\dagger A^T v $.
Below is the code for implementing the projection of a random vector onto the space of a random matrix. This is related to my other question Backward stable algorithm to get orthogonal projection onto the column space of a matrix. I didn't get an answer to that question.
I would like to know why is there a huge difference between the two methods of calculating the projection.
% testingprojfrostackexchange clear; M = 1400; N = 1300; r = 1; A = rand(M,N); u = rand(M,r); projLN = pinv(A')*(A'*u);%This is projection through Least Norm projLS = A*(pinv(A)*u);%This is projection through Least Square [Q R] = qr(A); Q = Q(:,1:N); z1 = Q*(Q'*u);%This is the actual projection display('(projection through QQT) - projLS'); norm(z1-projLN)/norm(projLN) display('(projection through QQT) - projLN'); norm(z1-projLS)/norm(projLS)
>> stackexchange (projection through QQT) - projLS ans = 2.1569e-13 (projection through QQT) - projLN ans = 8.3546e-15
The actual projection which is given by $Q(:,1:N)Q(:,1:N)^T$, where $Q$ is from the Housholder decomposition of $A$. We get the projections from $AA^\dagger$ and $(A^T)^\dagger A^T$ and find that $AA^\dagger$ is much better when $A$ is a tall matrix.