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reducible matrix (Definition)

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^TAP$ 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 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}\ne 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.



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Cross-references: information, Perron-Frobenius theorem, applications, index set, partition, strongly connected, digraph, equivalent, reducible, square matrix, upper triangular, block, permutation matrix, matrix
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This is version 8 of reducible matrix, born on 2002-12-22, modified 2007-10-26.
Object id is 3810, canonical name is ReducibleMatrix.
Accessed 12617 times total.

Classification:
AMS MSC: 15A48 (Linear and multilinear algebra; matrix theory :: Positive matrices and their generalizations; cones of matrices)
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