Let the adjacency matrix of an undirected bipartite graph be $A = \begin{pmatrix} 0 & B \\ B^T & 0 \end{pmatrix}$ where B is called the biadjacency matrix.

Now, by instead interpreting B as an adjacency matrix of an non-bipartite (arbitrary) directed graph, we get an equivalence relation between the bipartite undirected graph (with biadjacency matrix B) and the arbitrary directed graph (with adjacency matrix B).

Does this equivalence have a name? Is it discussed somewhere in literature? Thank you!