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We characterize sequences of positive integers $$(a_1,a_2,\ldots ,a_n)$$ ( a 1 , a 2 , … , a n ) for which the $$2\times 2$$ 2 × 2 matrix $$\left( \begin{array}{ll} a_n&{} \quad -\,1\\ 1&{}\quad 0 \end{array} \right) \left( \begin{array}{ll} a_{n-1}&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) \cdots \left( \begin{array}{ll} a_1&{}\quad -\,1\\ 1&{}\quad 0 \end{array} \right) $$ a n - 1 1 0 a n - 1 - 1 1 0 ⋯ a 1 - 1 1 0 is either the identity matrix $$\mathrm {Id}$$ Id , its negative $$-\,\mathrm {Id}$$ - Id , or square root of $$-\,\mathrm {Id}$$ - Id . This extends a theorem of Conway and Coxeter that classifies such solutions subject to a total positivity restriction.
Research in the Mathematical Sciences – Springer Journals
Published: Apr 9, 2018
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