TITLE: What is an "indecomposable" matrix?
QUESTION [6 upvotes]: 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?

REPLY [3 votes]: An $n \times n$ matrix $A$ is reducible if there is a permutation matrix $P$ so that
$$PAP^T=\begin{bmatrix}B&C\\0&D\end{bmatrix}$$
where $B$ and $D$ are square.
Equivalently, $A$ is reducible if it has a $k \times (n-k)$ zero submatrix with disjoint column and row indices.
If $A$ is not reducible, it is irreducible.
On the other hand, $A$ is partly decomposable if there are permutation matrices $P$ and $Q$ so that
$$PAQ=\begin{bmatrix}B&C\\0&D\end{bmatrix}$$
where $B$ and $D$ are square. Equivalently, $A$ is partly decomposable if it has a $k \times (n-k)$ zero submatrix.
If $A$ is not partly decomposable, it is fully indecomposable (or sometimes totally indecomposable).
Every fully indecomposable matrix is irreducible, but the converse is false in general.
Brualdi and Ryser's Combinatorial Matrix Theory provides the following comment on this terminology:

One may wonder why the adverbs "partly" and "fully" are being used
here. The reason is that the terminology "decomposable" and
"indecomposable" is sometimes used in place of "reducible" and
"irreducible." A reducible matrix of order $n$ also has a $k$ by $n-k$
zero submatrix for some integer $k$ but the row indices and column
indices of the zero submatrix are disjoint sets. Thus reducibility is
a more severe restriction on a matrix than that of part
decomposability.

Indecomposable almost always means irreducible rather than fully indecomposable, but most authors, as with Brualdi and Ryser above, prefer to stick to irreducible and fully indecomposable, avoiding indecomposable.