A square n×n matrix A = a_(i j) is called reducible if the indices 1, 2, ..., n can be divided into two disjoint nonempty sets i_1, i_2, ..., i_μ and j_1, j_2, ..., j_ν (with μ + ν = n) such that a_(i_α j_β) = 0 for α = 1, 2, ..., μ and β = 1, 2, ..., ν. 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.

square matrix