I've been dabbling with matrix computations in finite fields, and I've stumbled upon a pattern that I can't understand. Perhaps someone here could shed some light on it?
So, here's what's happening: In my application, after performing certain computations, I end up with a vector (like this one from $\mathbb{Z}_5$: $(2, 4, 4, 2, 1, 1)$), whose span is of interest in my problem. There are specific coordinates in this vector that are especially meaningful for what I'm working on. When I "normalize" this vector to make one of these significant coordinates 1, the other meaningful coordinates tend to fall into this neat pattern of being either 1 or -1, modulo the field's order.
For a specific example, consider the vector mentioned earlier, with the 3rd, 5th, and 6th coordinates being of particular importance for my application. By scaling this vector by 4 to adjust the 3rd coordinate to 1, it transforms into $(3,1,1,3,4,4)$. That is, the coordinates that matter for my application always end up being 1 or -1 after this "normalization".
I've noticed this happening consistently across different scenarios and field sizes, and can't figure why is this happening.
Let $\bar{x}\in\Bbb{F}_q^n$ be a vector in a finite-dimensional vector space over a finite field. Let $x_i\in\Bbb{F}_q$ be a 'meaningful' coordinate. If $x_i\neq0$ then we can scale the vector so that the 'meaningful' coordinate equals $1$, resulting in the vector $x_i^{-1}\bar{x}\in\Bbb{F}_q^n$. This tells us absolutely nothing about the other coordinates. Also note that this scaling is not possible if $x_i=0$.
So if it turns out that certain other coordinates always scale to $\pm1$, then you are selecting your vectors in a particular way to lead to this result. In particular, you must be selecting your vectors in such a way that the meaningful coordinates are nonzero, and in such a way that for every pair of meaningful coordinates $x_i$ and $x_j$ you have either $x_i=x_j$ or $x_i=-x_j$.