Consider the following equation $A_{n\times n}x_{n\times 1}=b_{n\times 1}$. This is a nonhomogeneous linear system, but the SVD is "specialized" for solving homogeneous systems, like $Ax=0$.
This Wikipedia article states that the mapping from the solution of a homogeneous system can be mapped to a solution of the nonhomogeneous system. So, if $v$ is the solution to $Ax=0$ and $p$ is any solution of $Ax=b$, then the entire solution can be described as $p+v$.
I'm confused about that description, as it indicates that I have to solve both systems in order to make that connection, which doesn't make sense.
So, can the linear system, depicted in its matrix form as $Ax=b$, be solved by applying the single value decomposition on its component $A$? If yes, how, if not, why?