A Property of Orthogonal Matrices

  • Alexander A. Roytvarf


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.


Vector Space Scalar Product Bilinear Form Orthogonal Projection Orthogonal Complement 


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© Springer Science+Business Media, LLC 2013

Authors and Affiliations

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

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