Incidence matrix and biadjacency matrix of bipartite graph

Question

For a simple bipartite graph, the adjacency matrix can be written as $$ A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix} $$ where $B$ is called biadjacency matrix. Is there a relation between the matrix $B$ and the unsigned incidence matrix $M_{+}$ and signed incidence matrix $M_{-}$. I know that I can write $$ 2 \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix} = \frac{1}{2} \left(M_{+}M_{+}^T - M_{-}M_{-}^T\right)$$ but was wondering if I can find $B$ as a function of $M_{+}$ and $M_{-}$ more explicitly. Thanks!