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)."
PlanetMath: reducible matrix