  In this paper, we prove that each matrix in $M_{m\times n}(\mathbb
Z_{\geq0})$ is uniformly column sign-coherent with respect to any $n\times n$
skew-symmetrizable integer matrix. Using such matrices, we introduce the
definition of irreducible skew-symmatrizable matrix. Based on this, the
existence of a maximal green sequence for a skew-symmetrizable matrices is
reduced to the existence of a maximal green sequence for irreducible
skew-symmetrizable matrices.
