  We develop a matrix approach to the Maximal Acyclic Subgraph (MAS) problem by
reducing it to finding the closest nilpotent matrix to the matrix of the graph.
Using recent results on the closest Schur stable systems and on minimising the
spectral radius over special sets of non-negative matrices we obtain an
algorithm for finding an approximate solution of MAS. Numerical results for
graphs from 50 to 1500 vertices demonstrate its fast convergence and give the
rate of approximation in most cases larger than 0.6. The same method gives the
precise solution for the following weakened version of MAS: and the minimal $r$
such that the graph can be made acyclic by cutting at most $r$ incoming edges
from each vertex. Several modifications, when each vertex is assigned with its
own maximal number $r_i$ of cut edges, when some of edges are "untouchable",
are also considered. Some applications are discussed.
