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A Property of Orthogonal Matrices

In: Thinking in Problems

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  • Alexander A. Roytvarf

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

Suggested Citation

  • Alexander A. Roytvarf, 2013. "A Property of Orthogonal Matrices," Springer Books, in: Thinking in Problems, edition 127, pages 107-137, Springer.
    • Handle: RePEc:spr:sprchp:978-0-8176-8406-8_8
    DOI: 10.1007/978-0-8176-8406-8_8
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