# reducible matrix

An $n\times n$ matrix $A$ is said to be a 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. 1.

$A$ is an irreducible matrix.

2. 2.

The digraph associated to $A$ is strongly connected.

3. 3.

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

4. 4.

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

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.

Title reducible matrix ReducibleMatrix 2013-03-22 13:18:20 2013-03-22 13:18:20 Mathprof (13753) Mathprof (13753) 11 Mathprof (13753) Definition msc 15A48 irreducible matrix