Suppose I have a system:

$$
Ax = b
$$

where $A$ is a $m$ by $n$ matrix which is less than full rank (neither full column nor row rank). In my particular case $m<n$.

I'd like a combination of a minimum norm solution (for elements of $x$ which are not determined and a least squares solution for those that are determined or over determined).

I know I can approximate this with Tikhonov regularization:

$$
\mathop{\text{minimize}}\limits_x \|Ax-b\|^2 + \alpha\|x\|^2.
$$

But this is not ideal as the regularizer tugs on all elements of $x$ not just those left undetermined. If I make $\alpha$ too large then my $x$ at determined or over determined elements will not be optimal (w.r.t. $\|Ax-b\|^2$). If $\alpha$ is too small, I imagine I run into conditioning issues.

Is there no way to formulate this correctly using some sort null space/QR decomposition?