Incidence matrix of a set system ($B_{i,j}=1_{i\in A_j}$)

For a finite ground set and an indexed set family, this zero-one matrix records membership of points in members. The diagonal entries of $B^{T}B$ are member sizes, and its off-diagonal entries count pairwise intersections. For the Fano plane, either point-line orientation gives the identity $B^{T}B=2I+J$, since every point lies on three lines and each distinct point pair lies on one line.