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A Property of Orthogonal Matrices

  • Alexander A. Roytvarf
Chapter

Abstract

Readers likely recall the following elementary geometric statement: if the sides of an angle α in a Euclidean plane are orthogonal to the sides of an angle β, then α=β or α+β=180°; in other words, |cos α|=|cos β|. A multidimensional generalization of this theorem relates to an interaction of skew symmetry and orthogonality features, i.e., properties of determinants and other multilinear functions, and Euclidean spaces; we presume readers are familiar with these features within the scope of a common university course, and on this basis we develop the tools necessary to prove this generalization. These tools have multiple applications. For examples, they allow us to extend a definition of the angle between straight lines or between hyperplanes to k-dimensional planes of ℝ n for every 0<k<n and to extend a definition of the vector product of two vectors in ℝ3−to any number of vectors and multivectors in ℝ n . Readers may learn more about the by consulting the guide to the literature provided in this chapter.

Keywords

Vector Space Scalar Product Bilinear Form Orthogonal Projection Orthogonal Complement 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

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Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  • Alexander A. Roytvarf
    • 1
  1. 1.Rishon LeZionIsrael

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