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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Products of reflections in the unitary group


Authors: Dragomir Ž. Djoković and Jerry Malzan
Journal: Proc. Amer. Math. Soc. 73 (1979), 157-160
MSC: Primary 22C05; Secondary 20G25
DOI: https://doi.org/10.1090/S0002-9939-1979-0516455-7
MathSciNet review: 516455
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Abstract: Let $ A \in U(n),\det (A) = \pm 1$ and let $ \exp (i{\alpha _k}),1 \leqslant k \leqslant n$ be the eigenvalues of A where $ 0 \leqslant {\alpha _1} \leqslant {\alpha _2} \leqslant \cdots \leqslant {\alpha _n} < 2\pi $. Then $ k(A) = ({\alpha _1} + \cdots + {\alpha _n})/\pi $ is an integer and $ 0 \leqslant k(A) \leqslant 2n - 1$. Denote by $ l(A)$ the length of A with respect to the set of all reflections, i.e., $ l(A)$ is the smallest integer m such that A is a product of m reflections. A reflection is a matrix conjugate to $ {\text{diag}}( - 1,1, \ldots ,1)$. Our main result is the formula $ l(A) = \max (k(A),k({A^\ast}))$.


References [Enhancements On Off] (What's this?)

  • [1] Heydar Radjavi, Decomposition of matrices into simple involutions, Linear Algebra and Appl. 12 (1975), no. 3, 247–255. MR 0414585

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1979-0516455-7
Keywords: Unitary group, reflections, length, eigenvalues, angles, characteristic polynomial
Article copyright: © Copyright 1979 American Mathematical Society