Angles Between Subspaces Computed in Clifford Algebra

Hitzer, Eckhard

doi:10.1063/1.3498043

Cited by 8 publications

(7 citation statements)

References 5 publications

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“…For the product of a vector and a blade we have Remark 2.7. An alternative proof can be achieved from looking at the geometric product AB as detailed in [19] equation (45) and more generally in [20] equation (17). There the lowest order term is a product of all the cosines of the principal angles and the lowest +2 order term is a sum of summands each with the product of one sine of one principal angle times the cosines of all other principal angels.…”

Section: Coorthogonality and Basesmentioning

confidence: 99%

A General Geometric Fourier Transform Convolution Theorem

Bujack

Scheuermann

Hitzer

2012

Adv. Appl. Clifford Algebras

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Abstract. The large variety of Fourier transforms in geometric algebras inspired the straight forward definition of "A General Geometric Fourier Transform" in Bujack et al., Proc. of ICCA9, covering most versions in the literature. We showed which constraints are additionally necessary to obtain certain features like linearity, a scaling, or a shift theorem. In this paper we extend the former results by a convolution theorem.

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Section: Coorthogonality and Basesmentioning

confidence: 99%

A General Geometric Fourier Transform Convolution Theorem

Bujack

Scheuermann

Hitzer

2012

Adv. Appl. Clifford Algebras

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“…There is also a number of concepts, like p-dimensional angle [Jia96], total angle [Hit10], product angle [GH06], and others [Afr57, Glu67, MBI96, Wed83], which are closely related to one another. In section 3 we define a Grassmann angle which unifies and extends these.…”

Section: Angles Between Subspacesmentioning

confidence: 99%

“…Many distinct concepts of angle between subspaces can be found in the literature [Dix49,Fri37,Glu67,Jor75,Wed83], having applications in geometry, linear algebra, and other areas as diverse as functional analysis [KJA10], statistics [Hot36] and data mining [JSLG18]. Some recent works on the subject can be found in [BT09,GH06,GNSB05,Hit10].…”

Section: Introductionmentioning

confidence: 99%

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Grassmann angles between real or complex subspaces

Mandolesi

2019

Preprint

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The Grassmann angle unifies and extends several concepts of angle between subspaces found in the literature, being defined for subspaces of equal or different dimensions, in a real or complex inner product space. It is related to Grassmann's exterior algebra and to the Fubini-Study and Hausdorff distances, and describes how Lebesgue measures contract under orthogonal projections. This angle is asymmetric for subspaces of different dimensions, turning the total Grassmannian of all subspaces into a quasipseudometric space. Other odd characteristics, already present in former angle concepts but usually overlooked, are explained, like its behavior with respect to orthogonal complements and realifications. We use it to study the Plücker embedding of Grassmannians, and to get an obstruction on complex structures for pairs of subspaces.

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“…The Grassmann angle between subspaces was introduced in [14] to unify similar angles found in the literature [6,8,9,10,12,17], and extend them for arbitrary dimensions and complex spaces. It has many useful properties which these other angles do not, like, for example, a triangle inequality which holds for subspaces of different dimensions.…”

Section: Introductionmentioning

confidence: 99%

Grassmann angle formulas and identities

Mandolesi

2020

Preprint

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Grassmann angles between real or complex subspaces measure the contraction of volumes in orthogonal projections. Their relations with contractions, inner and exterior products of simple multivectors are used to obtain formulas for computing Grassmann and complementary Grassmann angles in terms of arbitrary bases of the subspaces. We also obtain several identities for Grassmann angles with certain families of subspaces, including generalizations of the Pythagorean identity cos 2 θ + sin 2 θ = 1 for higher dimensions and for complex spaces, which can be used to get generalized Pythagorean theorems. These identities have interesting connections with quantum theory and Clifford's geometric algebra.

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Angles Between Subspaces Computed in Clifford Algebra

Cited by 8 publications

References 5 publications

A General Geometric Fourier Transform Convolution Theorem

A General Geometric Fourier Transform Convolution Theorem

Grassmann angles between real or complex subspaces

Grassmann angle formulas and identities

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