Let*M* be a square matrix whose entries are in some field. Our object is to find a permutation matrix*P* such that*PM P*^{−1} is completely reduced, i.e., is partitioned in block triangular form, so that all submatrices below its diagonal are 0 and all diagonal submatrices are square and irreducible. Let*A* be the binary (0, 1) matrix obtained from*M* by preserving the 0's of*M* and replacing the nonzero entries of*M* by 1's. Then*A* may be regarded as the adjacency matrix of a directed graph*D*. Call*D* strongly connected or*strong* if any two points of*D* are mutually reachable by directed paths. A*strong component* of*D* is a maximal strong subgraph. The*condensation D*^{*} of*D* is that digraph whose points are the strong components of*D* and whose lines are induced by those of*D*. By known methods, we construct*D*^{*} from the digraph,*D* whose adjacency matrix*A* was obtained from the original matrix*M*. Let*A*^{*} be the adjacency matrix of*D*^{*}. It is easy to show that there exists a permutation matrix*Q* such that*QA*^{*}*Q*^{−1} is an upper triangular matrix. The determination of an appropriate permutation matrix*P* from this matrix*Q* is straightforward.