On the decomposition of orthogonalities into symmetries

“…Every good geometry book proves that each isometry of euclidean nspace is a product of at most n+1 reflections and several more-advanced sources include Scherk's theorem which identifies the minimal length of such a reflection factorization from the basic geometric attributes of the isometry under consideration [Sch50,Die71,ST89,Tay92]. The structure of the full set of minimal length reflection factorizations, on the other hand, does not appear to have been given an elementary treatment in the literature even though the proof only requires basic geometric tools.…”

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confidence: 99%

Factoring Euclidean isometries

Brady

McCammond

2015

Int. J. Algebra Comput.

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Abstract. Every isometry of a finite dimensional euclidean space is a product of reflections and the minimum length of a reflection factorization defines a metric on its full isometry group. In this article we identify the structure of intervals in this metric space by constructing, for each isometry, an explicit combinatorial model encoding all of its minimal length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved.Every good geometry book proves that each isometry of euclidean nspace is a product of at most n+1 reflections and several more-advanced sources include Scherk's theorem which identifies the minimal length of such a reflection factorization from the basic geometric attributes of the isometry under consideration [Sch50, Die71, ST89, Tay92]. The structure of the full set of minimal length reflection factorizations, on the other hand, does not appear to have been given an elementary treatment in the literature even though the proof only requires basic geometric tools.1 In this article we construct, for each isometry, an explicit combinatorial model encoding all of its minimal length reflection factorizations. The model is largely independent of the isometry chosen in that it only depends on whether or not some point is fixed and the dimension of the space of directions that points are moved.Analogous results for spherical isometries already exist and are easy to state: when w is an orthogonal linear transformation of R n only fixing the origin, for example, there is a natural bijection between minimal length factorizations of w into reflections fixing the origin and complete flags of linear subspaces in R n [BW02]. In other words, the structure

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“…In 1948 J. Dieudonné extended the result to arbitrary fields with characteristic not 2 [3]. Finally, P. Scherk obtained the minimal number and dealt with arbitrary fields F of char F ̸ = 2 in 1950 [11]. In the literature the CDS Theorem is often stated with the non-minimal upper bound of n factors and Scherk's name is excluded; see [4,12].…”

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confidence: 99%

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confidence: 99%

“…Scherk's proof [11] uses constructions in parts, but some key steps are proved through existence arguments. Specifically, the existence of a non-isotropic vector that is needed to form a Householder factor [11,Lemma 4] is not proved in a constructive way by Scherk. The outline of our proof closely resembles that of [11] restricted to F equal to R or C, but we construct all elements for the S-Householder factorization of a generalized orthogonal matrix explicitly rather than relying on their existence.…”

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confidence: 99%

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A constructive proof of the Cartan–Dieudonné–Scherk Theorem in the real or complex case

Fuller

2011

Journal of Pure and Applied Algebra

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a b s t r a c tThe Cartan-Dieudonné-Scherk Theorem states that for fields of characteristic other than 2, every orthogonality can be written as the product of a certain minimal number of reflections across hyperplanes. The earliest proofs are not constructive, and constructive proofs either do not achieve minimal results or have been restricted to special cases. This paper presents a constructive proof in the real or complex field of the decomposition of a generalized orthogonal matrix into the product of the minimal number of generalized Householder matrices.

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On the decomposition of orthogonalities into symmetries

Cited by 53 publications

References 0 publications

A Partial Order on the Orthogonal Group

A Partial Order on the Orthogonal Group

Factoring Euclidean isometries

A constructive proof of the Cartan–Dieudonné–Scherk Theorem in the real or complex case

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