Cauchy-binet for Pseudo-determinants

The pseudo-determinant Det(A) of a square matrix A is defined as the product of the nonzero eigenvalues of A. It is a basis-independent number which is up to a sign the first nonzero entry of the characteristic polynomial of A. We prove Det(FG) = ∑ P det(FP)det(GP) for any two n×m matrices F,G. The sum to the right runs over all k × k minors of A, where k is determined by F and G. If F = G is the incidence matrix of a graph this directly implies the Kirchhoff tree theorem as L = FG is then the Laplacian and det(FP ) ∈ {0, 1} is equal to 1 if P is a rooted spanning tree. A consequence is the following Pythagorean theorem: for any self-adjoint matrix A of rank k, one has Det(A) = ∑ P det (AP), where det(AP ) runs over k × k minors of A. More generally, we prove the polynomial identity det(1 + xFG) = ∑ P x |P det(FP )det(GP ) for classical determinants det, which holds for any two n×m matrices F,G and where the sum on the right is taken over all minors P , understanding the sum to be 1 if |P | = 0. It implies the Pythagorean identity det(1 +FF ) = ∑ P det (FP ) which holds for any n ×m matrix F and sums again over all minors FP . If applied to the incidence matrix F of a finite simple graph, it produces the Chebotarev-Shamis forest theorem telling that det(1 + L) is the number of rooted spanning forests in the graph with Laplacian L.