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.
A square matrix that is not reducible is said to be irreducible.
