  A square (0,1)-matrix X of order n > 0 is called fully indecomposable if
there exists no integer k with 0 < k < n, such that X has a k by n-k zero
submatrix. A stable set of a graph G is a subset of pairwise nonadjacent
vertices. The stability number of G, denoted by $\alpha (G)$, is the
cardinality of a maximum stable set in G. A graph is called $\alpha $-stable if
its stability number remains the same upon both the deletion and the addition
of any edge. We show that a connected bipartite graph has exactly two maximum
stable sets that partition its vertex set if and only if its reduced adjacency
matrix is fully indecomposable. We also describe a decomposition structure of
$\alpha $-stable bipartite graphs in terms of their reduced adjacency matrices.
On the base of these findings we obtain both new proofs for a number of
well-known theorems on the structure of matrices due to Brualdi, Marcus and
Minc, Dulmage and Mendelsohn, and some generalizations of these statements.
Several new results on $\alpha $-stable bipartite graphs and their
corresponding reduced adjacency matrices are presented, as well. Two kinds of
matrix product are also considered (namely, Boolean product and Kronecker
product), and their corresponding graph operations. As a consequence, we obtain
a strengthening of one Lewin's theorem claiming that the product of two fully
indecomposable matrices is a fully indecomposable matrix.
