Acta Mathematica Vietnamica
Products of Reflections and Commutators of Reflections Over Division Algebras
M. H. Bien
,
N. T. T. Ha
,
T. N. N. Hung
,
D. T. Toan
Abstract
Let $D$ be a division algebra and let $n$ be a positive integer greater than 1. Assume that the commutator width $\omega (D)$ of $D^{*}$ is positive and finite. First, we revisit a question posed by F. S. Cater concerning the decomposition of matrices into a product of reflections over a division ring. Among results, we show that every matrix $A$ in the special linear group $\textrm{SL}_n(D)$ can be expressed as a product of at most $\textrm{rank} (A-\mathrm I_n)+4\omega (D)$ reflections. Next, we study the decomposition of matrices in $\textrm{SL}_n(D)$ into products of commutators of reflections in the general linear group $\textrm{GL}_n(D)$.
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