reducible matrix

An $n\times n$ matrix $A$ is said to be a reducible matrix if and only if for some permutation matrix $P$, the matrix $P^{T}AP$ is block upper triangular.If a square matrix is not reducible, it is said to be an irreducible matrix.

The following conditions on an $n\times n$ matrix $A$ are equivalent.

1. $A$ is an irreducible matrix.

2. The digraph associated to $A$ is strongly connected.

3. For each $i$ and $j$, there exists some $k$ such that ${(A^k)}_{ij}>0$.

4. For any partition $J\bigsqcup K$ of the index set $\{1,2,\mathrm{\dots },n\}$, there exist $j\in J$ and $k\in K$such that $a_{jk}\ne 0$.

For certain applications, irreducible matrices are more useful than reducible matrices. In particular, the Perron-Frobenius theorem gives more information about the spectra of irreducible matrices than of reducible matrices.