Solution of under determinant system of equations

Question

What is the mathematical technique to solve homogeneous under determined system of eqns. Ax=0, while minimizing the vector x in the sense of L1 norm?

Answer 1

Solving $Ax=0$ means finding the basis of null space of $A$.

One way of finding the basis of null space is to compute SVD of A: $$A = U\Sigma V^*$$

The right-singular vectors ($V$) corresponding to vanishing singular values ($\Sigma$) of $A$ span the null space of $A$. Now, given the whole basis of null space, find the columns which has minimum sum and that is your solution.

Reference: https://en.wikipedia.org/wiki/Singular-value_decomposition#Range,_null_space_and_rank