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) = P det(F P )det(G P ) 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 = F T G is then the Laplacian and det 2 (F P ) ∈ {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) = P det 2 (A P ), where det(A P ) runs over k × k minors of A. More generally, we prove the polynomial identity det(1 + xF T G) = P x |P | det(F P )det(G P ) 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 + F T F ) = P det 2 (F P ) which holds for any n × 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.where the sum is over all m × m square sub matrices P and F P is the matrix F masked by the pattern P . In other words, F P is an m × m matrix obtained by deleting n − m rows in F and det(F P ) is a minor Date: June 17, 2014. 1991

SMASH: Sparse Matrix Atomic Scratchpad Hashing

“…In 1812, a French mathematician named Jacques Philippe Marie Binet pointed out several important computations involved the multiplication of two matrices [53]. On November 30 of the same year, he provided a lecture on his observation and further extended his work, leading to the Cauchy-Binet formula [54]. This is one of the oldest known sources of the discovery of matrix multiplication.…”

Effective metrics and a fully covariant description of constitutive tensors in electrodynamics

“…Now we are in the position to actually generalize the det ǫ ij or det µ −1 ij terms to a fully covariant formalism 9 [25] also contains some more historic references about other (re)discoveries of the pseudo-inverse. 10 Early notions of the pseudo-determinant can be found in [26], while more modern appearances include [27,28]. Written as det ′ (A), a similar notion for operators can be found in the quantum field theory literature in [29] and probably even earlier.…”

Differentiating the pseudo determinant

“…See [1] for an equivalent definition of the pseudo determinant in terms of the characteristic polynomial. In deriving its derivative, it will be useful to write the pseudo determinant as a limit.…”

“…The pseudo determinant arises in graph theory within Kirchoff's matrix tree theorem [1] and in statistics, in the definition of the degenerate Gaussian distribution. The degenerate Gaussian has been useful in image segmentation [2], communications [3], and as the asymptotic distribution for multinomial samples [4].…”