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# 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 Perron-Frobenius theorem gives more information about the spectra of irreducible matrices than of reducible matrices.

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irreducible matrix

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Definition

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## Mathematics Subject Classification

15A48*no label found*

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