Abstract
In this chapter the structure of the orthogonal group is studied in more depth. In particular, we prove that every isometry in O(n) is the composition of at most n reflections about hyperplanes (for n ≥ 2, see Theorem 8.1). This important result is a special case of the “Cartan–Dieudonn’e theorem” (Cartan [4], Dieudonn’e [6]). We also prove that every rotation in SO(n) is the composition of at most n flips (for n ≥ 3).
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Gallier, J. (2011). The Cartan–Dieudonné Theorem. In: Geometric Methods and Applications. Texts in Applied Mathematics, vol 38. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-9961-0_8
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DOI: https://doi.org/10.1007/978-1-4419-9961-0_8
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