Products of Reflections and Triangularization of Bilinear Forms

Jacques Helmstetter


The present article is motivated by the theorem of Cartan-Dieudonn\'e which states that every orthogonal transformation is a product of reflections. Its purpose is to determine, for each orthogonal transformation, the minimal number of factors in a decomposition into a product of reflections, and to propose an effective algorithm giving such a decomposition. With the orthogonal transformations $g$ of a quadratic space $(V,q)$, it associates couples $(S,\phi)$ where $S$ is a subspace of $V$, and $\phi$ an non-degenerate bilinear form on $S$ such that $\phi(y,y)=q(y)$ for every $y$ in $S$. In general, the minimal decompositions of $g$ into a product of reflections correspond to the bases of $S$ in which the matrix of $\phi$ is lower triangular. Therefore, we need an algorithm of triangularization of bilinear forms. Affine isometries are also taken into consideration.

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Journal of Mathematics Research   ISSN 1916-9795 (Print)   ISSN 1916-9809 (Online)

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