Let $B$ be an $m$ by $n$ matrix whose entries are either 1 or 0 (it is an undirected incidence matrix). Given the $n$ by $n$ matrix $Q$, defined by $B^{\text{T}}B=Q$, is it possible to solve for $B$? If so, how? And is the solution unique?
The application here is, if I know an adjacency matrix of an undirected graph, can I get the incidence matrix?
Let $B$ and $C$ be the respective incidence matrices of $K_{1,3}$ and of $K_3$ with an extra isolated vertex. Then $B^TB$ and $C^TC$ both equal $I+J$, where $J$ is the all-ones matrix. But $B$ and $C$ are not permutation equivalent.