What is an "indecomposable" matrix?

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.

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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.