I'm reading an article where, at some point, the following Identity is used:
$$|\det(C^{+}-C^{-})_{S}| =|\sum_{I\subset S_{1}}\sum_{J\subset S_{2}}\pm\det C^{+}_{I\times J}\det C^{-}_{(S_{1}\setminus I)\times (S_{2} \times J)}|$$ where $C=C^{+}-C^{-} $ is an $n \times n $ matrix with $C^{\pm} $ both positive-definite and non-degenerate, $S=S_{1}\cup S_{2} $ is an union of disjoint even sets and $C_{S} $ denotes the submatrix $(c_{ij})_{i,j \in S} $ . Now, does anyone know what identity is this? It seems some sort of Cauchy-Binet formula but for a sum rather than a product of matrices. How to prove it?