A multilinear algebra proof of the Cauchy–Binet formula and a multilinear version of Parsevalʼs identity

Konstantopoulos, Takis

doi:10.1016/j.laa.2013.07.009

Cited by 3 publications

(3 citation statements)

References 14 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Remarks. 1) Despite the simplicity of the proof and similar looking results for minor expansion, formulas in multi-linear algebra [33,4], condensation formulas, trace ideals [51], matrix tree results [7], formulas for the characteristic polynomial [37,45,11], pseudo inverses, non-commutative generalizations [49], we are not aware that even the special case of Corollary 6) for the pseudo determinant has appeared anywhere already. In the classical case, where A is invertible, the Pythagorean identity is also called Lagrange Identity [4].…”

Section: Theorem 7) Impliesmentioning

confidence: 99%

Cauchy–Binet for pseudo-determinants

Knill

2014

Linear Algebra and its Applications

View full text Add to dashboard Cite

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

show abstract

Section: Theorem 7) Impliesmentioning

confidence: 99%

Cauchy–Binet for pseudo-determinants

Knill

2014

Linear Algebra and its Applications

View full text Add to dashboard Cite

show abstract

“…Furthermore, applying the formula for the generalized expansion of a determinant, we transform the determinant in (13) into the…”

Section: Resultsmentioning

confidence: 99%

“…The Cauchy-Binet formula plays an important role in studies of determinants, permanents and other classes of matrix functions. An increasing interest in its applications in many branches of applied science, such as matrix analysis and engineering [11][12][13], is a motivation of the paper. It is worth emphasizing that, so far, many considerable contributions to generalizing the Cauchy-Binet theorem have been made [14][15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Cauchy-Binet type formulas for Fredholm operators

Ciecierska

2017

J. Appl. Math. Comput. Mech.

View full text Add to dashboard Cite

are Fredholm operators acting in linear spaces. By referring to the correspondence between Fredholm operators and their determinant systems, we derive the formulas for a determinant system for AB which are expressed via determinant systems for A and B. In our approach, applying results of the theory of determinant systems plays the crucial role and yields Cauchy-Binet type formulas. The formulas are utilized in many branches of applied science and engineering.MSC 2010: 47A53, 15A15

show abstract