A vector is $k$-sparse, if there are at most $k$ non-zero entries.
In compressed sensing an arbitrary matrix $A\in\mathbb{R}^{m\times n}$ (with usually $m<n$) is said to have the null space property (for non-negative signal) for $k$, if every $k$-sparse non-negative vector $x^*$ is the unique positive solution of $Ax=Ax^*$.
This is equivalent to: Every non-zero null space vector of $A$ has at least $k+1$ negative entries.
I am just searching for an example for a matrix $A$ with this nullspace property for small $k>0$. Does anybody know one?
The matrix $$\pmatrix{1&0&1&0\cr0&1&1&0\cr0&1&0&1\cr}$$ has nullspace generated by $(1,1,-1,-1)$, so every nonzero vector in the nullspace has at least 2 negative entries.