The number of connected components in a graph associated with a rectangular (0,1)(0,1)-matrix

With a nonzero rectangular (0,1)(0,1)-matrix A   we associate an undirected graph GAGA that corresponds to the linear transformation X↦AXTAX↦AXTA. We use the generalized singular-value decomopsition of incidence matrices to count the number of connected components and the number of bipartite connected components of GAGA. We show that GAGA is connected if and only if GAGA has no bipartite connected component or isolated vertex. We also show that if A   has no row or column of 0's, then the number of connected components is 12s(s+1) and the number of bipartite connected components is 12s(s−1), where s is the number of chainable components of A, that is, the number of connected components in the undirected graph with adjacency matrix(OAATO).