2013
DOI: 10.1016/j.laa.2013.07.009
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A multilinear algebra proof of the Cauchy–Binet formula and a multilinear version of Parsevalʼs identity

Abstract: We give a short proof of the Cauchy-Binet determinantal formula using multilinear algebra by first generalizing it to an identity not involving determinants. By extending the formula to abstract Hilbert spaces we obtain, as a corollary, a generalization of the classical Parseval identity.

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Cited by 3 publications
(3 citation statements)
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“…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%
“…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%
“…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%