How does one show a matrix is irreducible and reducible?

How does one show a matrix is irreducible and reducible? Please explain and an example would be great as well.

I know that a matrix is reducible if and only if it can be placed into block upper-triangular form. How do you find block upper-triangular form?

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Maybe you can remind us --- what does irreducible mean, in the context of matrices? what does reducible mean? –  Gerry Myerson Feb 27 '13 at 2:01

A square matrix is reducible iff the associated directed graph has smaller strongly connected components. So you may use a strong component algorithm to solve your problem.

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Please what is an associated directed graph of a matrix? –  npisinp Apr 11 '14 at 14:04
@npisinp Let the matrix in question be $A$ and let $B=(b_{ij})$ be the matrix such that $b_{ij}=1$ if $a_{ij}\ne0$, and $b_{ij}=0$ if $a_{ij}=0$. Then $B$ is the adjacency matrix of a directed graph, and $A$ is reducible iff this directed graph has proper strongly connected components. –  user1551 Apr 13 '14 at 12:23

The best place to look is this wiki link. To add to the other answer, another equivalent condition is that for every index $[i,j]$, there should be a $m$ such that $(A^m)_{ij}>0$ which is naturally satisfied if the matrix entries are all positive. If it is non-negative, then one needs to check other things.

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In my opinion this answer is a bit broad, because the provided wiki link links to a theorem that uses the irreducibilty of a Matrix. That is, there is no algorithm outlined to solve the reduciblity question. –  kiltek Dec 3 '14 at 9:46