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 certain applications, irreducible matrices are more useful than reducible matrices. In particular, the PerronFrobenius theorem^{} gives more information about the spectra of irreducible matrices than of reducible matrices.
Title  reducible matrix 

Canonical name  ReducibleMatrix 
Date of creation  20130322 13:18:20 
Last modified on  20130322 13:18:20 
Owner  Mathprof (13753) 
Last modified by  Mathprof (13753) 
Numerical id  11 
Author  Mathprof (13753) 
Entry type  Definition 
Classification  msc 15A48 
Defines  irreducible matrix 