How do you solve a least square problem with a noninvertible matrix? How do you find a solution to a matrix $A$ that minimizes $\|x\|$ when $A^TA$ is  not invertible? The matrix is $$A = \pmatrix{1 &1&2&2\\1&2&3&4}$$
I don't know if this helps but also in the question above this one, we are asked to find all solutions to $Ax = \pmatrix{0\\11}$
Thank you.
 A: As others have assumed, I am assuming that this problem is linked to the previous one and that we are looking to minimize $\|x\|$ where $Ax=\pmatrix{0\\11}$ and $A = \pmatrix{1&1&2&2\\1&2&3&4}$. To minimize $\|x\|$, we can minimize $\|x\|^2=x^Tx$. To minimize $x^Tx$ over all $x$ so that $Ax=\pmatrix{0\\11}$, $x^T$ must be in the row space of $A$.
Suppose $AA^Tu=\pmatrix{0\\11}$. Then, it is simple to show that $\|A^Tu-x\|^2=\|x\|^2-u^T\pmatrix{0\\11}$, and from there, it is easy to show that $x=A^Tu$ minimizes $\|x\|$.
If $AA^T$ is invertible, then you can find such a $u$.
Pseudoinverses:
It should be mentioned that when $AA^T$ is invertible, $A^T(AA^T)^{-1}$ is called the Moore-Penrose Pseudoinverse, or simply the pseudoinverse.
Mathematica:

A: I suppose that you are looking to find the value of $x$ for which $(Ax−b)^\intercal(Ax−b)$ attains the minimum. As you said this problem cannot be solved as $A$ is noninvertible and I cannot see how there can be a unique solution unless we impose additional constraints on $x$.
