Cauchy-Binet for pseudo determinants

Abstract. 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(FTG) =∑ 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 = FTG is then the Laplacian and det2(FP) ∈ {0, 1} is equal to 1 if P is a rooted spanning tree. A consequence is the following Pythagorean theo-rem: for any self-adjoint matrix A of rank k, one has Det2(A) =∑ P det 2(AP), where det(AP) runs over k × k minors of A. More generally, we prove the polynomial identity det(1 + xFTG) =∑ 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 +FTF) = P de