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What is an indecomposable matrix? I tried to find what it is, but Wikipedia does not have an entry for it. Also, what properties does such matrix have?

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You should provide some background. In what context did you find this term? – Giuseppe Negro Dec 16 '12 at 2:08
up vote 3 down vote accepted

See this entry in Planet Math: Fully Indecomposable Matrix.

See also Special matrices: scroll to "Decomposable". You'll find what it means to be decomposable, partly decomposable, and fully indecomposable.

Note: The term irreducible is usually used instead of indecomposable.

Wikipedia: "...a matrix is irreducible if it is not similar via a permutation to a block upper triangular matrix (that has more than one block of positive size)."
(Replacing non-zero entries in the matrix by one, and viewing the matrix as the adjacency matrix of a directed graph, the matrix is irreducible if and only if the digraph is irreducible.)

PlanetMath: 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 $P^TAP$ is block upper triangular matrix."
If a square matrix is not reducible, it is said to be an irreducible matrix.

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I think "irreducible" is much, much popular than "indecomposable". – user1551 Dec 16 '12 at 2:35
I agree, @user1551 ! – amWhy Dec 16 '12 at 2:35
+1 hope you are well, Amy. :) – S. Snape Aug 17 '13 at 3:35

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