A square matrix is called reducible if the indices 1, 2, ..., can be divided into two disjoint nonempty
sets ,
, ..., and , , ..., (with ) such that

for ,
2, ...,
and ,
2, ..., .

A matrix is reducible if and only if it can be placed into block upper-triangular form by simultaneous row/column permutations. In addition, a matrix is reducible if and only if its associated digraph is not strongly connected.