Matrix Null Vector with L1 norm

I have a matrix A (tall matrix N×27N×27, NN ~ 10001000). I am wanting to find its null space (Ax=0Ax=0) , thus I am trying to –

minimize_x ||Ax||_2

I am aware that under L2 norm I will have to enforce an additional constraint for example x27=1x27=1 or ||x||2=1||x||2=1. If using the later constraint a solution can be found by use of SVD.

However, I am wanting to find the null space under L1 norm, thus my minimization problem become

My question is what are some of the standard additional constraints that are in use to enable be to solve the L1 norm.

Do share any additional incite that you may have on this.

As I’m sure you’re aware a constraint such as \|x\|=1 is nonconvex and cannot be used with CVX. We don’t give general modeling advice on this forum so I suggest you take this to a more general forum like Math StackExchange.