Let's say $A$ is an $m\times n$ matrix and let's consider first that $A$ is of full rank that is the rank of $A$ is $r=\min(m,n)$
The goal is to solve the least square problem $Ax=0$
Define the error $E(x)=\|Ax\|_{2}^{2}$.
Since $A$ is of full rank, why can't we just say that $x=0$ is the solution to this homogenous least square problem? why is this not preferred over the usual solution to this problem that is the eigenvector corresponding the smallest eigenvalue of $A^{T}A$?
Alternatively, if $r<\min(m,n)$ then we may have infinite solution or no solution right?