What is an "indecomposable" matrix? 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?
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.
An irreducible matrix has a corresponding graph which is strongly connected. On the other hand, an indecomposable matrix has a corresponding graph which has a directed spanning tree. So irreducibility is a stronger condition than indecomposibility.