Let $V$ be a 3-dimensional vector space over a finite field $F$ of $q$ elements, where $q$ is an odd prime power. We know that the orthogonal group $O(V)$ is generated by reflections. How can a given element of $O(V)$ (in particular of $SO(V)$) be written explicitly as a product of reflections? For example, let $e,f,d$ be a standard basis for $V$, where $(e,e)=(f,f)=(e,d)=(e,f)=0$, $(e,f)=1$, $(d,d)=1$, and let $g\in SO(V)$ be the element which sends $e\to -e$, $f\to -f$, $d\to d+e$.
What would be the procedure of expressing such an element as a product of reflections? I have tried combining some randoms reflections together, but I haven't been able to find an expression for $g$, so I was wondering if there is a method to do this.
Thanks in advance for any help!
One answer is "first find a 2D invariant subspace". Typically you'd do thus by looking at the characteristic polynomial, factoring over the reals, and each quadratic factor will have a pair of conjugate eigenvalues, and these will determined a 2-dimensional (or 2k-dimensional) subspace.
Then you solve the problem on that 2D subspace, and then recursively on the complementary $n-2$-dimensional subspace.
For a 2D rotation, to rotate $e_1$ to $v = (\cos \theta, \sin \theta)$ via two reflections, you can reflect about the axis at angle $\theta/2$, and then reflect about $v$. This takes $e_1$ to $v$, and then $v$ to $v$, and hence the composition takes $e_1$ to $v$. And since it's a rotation of the plane, it has to be the right one.
Alternatively, and probably more simply, you just look up a description of using Householder reflections to do a QR factorization of your matrix. Trefethen and Bao's Numerical Linear Algebra has a nice description in "Lecture 10".