# Minimum norm solution to a (fake) overdetermined system which is actually underdetermined

Given a system of linear equations in $$n$$-vector $$x$$

$$Ax=b$$

where $$A$$ is an $$m \times n$$ matrix and $$b$$ is an $$m$$-vector. The system is

• overdetermined if $$n>m$$ (more variables than linear constraints). The least square solution is well-defined and coded in many solvers.

• underdetermined if $$n. The minimum norm solution is well-defined and coded in many solvers.

How to manipulate the solution if an overdetermined system turns to be underdetermined? In other words, some rows of $$A$$ are linearly dependent and there are actually less constraints than variables, though $$n>m$$ holds.

I guess SVD reflects whether the system is over or under determined. But what is the simple method to compute the minimum norm solution to such fake overdetermined system? Do commercial solvers automatically give such solutions?

• Why don't you start with Gaussian elimination? Find out if there even is a solution. If so, parametrize the set of solutions. Then go from there. – Rodrigo de Azevedo May 31 at 7:02