Ivan Shishkin, Rye (1878)

Problems/Linear algebraReviewed

The set of orthogonal matrices contains only trivial intervals

by Sequoia·
67
Difficulty scaleÉchelle de difficulté

This score reflects both the level of the required concepts and the difficulty of the solution.Ce score tient compte à la fois du niveau des notions nécessaires et de la difficulté de la résolution.

  1. 1–10First steps / middle schoolPremiers pas / collège
  2. 11–25Beginner / high schoolDébutant / lycée
  3. 26–50Intermediate / undergraduateIntermédiaire / licence
  4. 51–70Advanced / graduateAvancé / master
  5. 71–90Expert / specializedExpert / spécialisé
  6. 91–100Research levelNiveau recherche
These levels are approximate guides.Ces niveaux sont des repères approximatifs.
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English
EnglishAdd translation

Let A,B∈On(R)A,B\in\mathcal{O}_{n}(\R) be two orthogonal matrices.

Give a necessary and sufficient condition that must hold for these matrices so that A+B2\frac{A+B}{2} is also orthogonal.

Conclude that On(R)\mathcal{O}_{n}(\R) contains no nontrivial intervals, hence it is not convex.

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