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.