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Journal Article

# Products of Reflections in the Unitary Group

Dragomir Ž. Djoković and Jerry Malzan
Proceedings of the American Mathematical Society
Vol. 73, No. 2 (Feb., 1979), pp. 157-160
DOI: 10.2307/2042282
Stable URL: http://www.jstor.org/stable/2042282
Page Count: 4

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Topics: Eigenvalues, Geometric angles, Mathematical theorems, Integers, Matrices, Unit vectors

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Let $A \in U(n), \det(A) = \pm1$ 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 $\operatorname{diag}(-1, 1,\ldots, 1)$. Our main result is the formula $l(A) = \max(k(A), k(A^\ast))$.