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(F^T G) = sum_P det(F_P) det(G_P) for any two n times m matrices F,G. The sum to the right runs over all k times 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=F^T G is then the Laplacian and det^2(F_P) in {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^2(A) = sum_P det^2(A_P), where det(A_P) runs over k times k minors of A. More generally, we prove the polynomial identity det(1+x F^T G) = sum_P x^{|P|} det(F_P) det(G_P) for classical determinants det, which holds for any two n times 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+F^T F) = sum_P det^2(F_P) which holds for any n times m matrix F and sums again over all minors F_P. 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.