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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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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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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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