MHB Solving for Distinct Vectors of G: $\mathbb{F}_p$, n, and v

  • Thread starter Thread starter jakncoke1
  • Start date Start date
  • Tags Tags
    Vectors
AI Thread Summary
The discussion focuses on determining pairs of p and n for which a fixed vector v and a matrix M can be defined such that the k-fold composition of the function G produces distinct vectors in $\mathbb{F}_p^{n}$. The function G is defined as G(x) = v + Mx, and the goal is to ensure that the outputs G^k(0) for k ranging from 1 to p^n are all unique. The problem requires a deep understanding of linear transformations and modular arithmetic. Participants are encouraged to share insights and strategies for approaching the problem, emphasizing the importance of considering the properties of the matrix M and the vector v. The thread highlights the complexity of achieving distinct outputs in the context of finite fields.
jakncoke1
Messages
48
Reaction score
1
Let $\mathbb{F}_p$ denote the integers mod p and let n be a positive integer.
Let v be a fixed vector $\in \mathbb{F}_p^{n}$, Let M be a nxn matrix with entries from $\mathbb{F}_p$. Define G:$\mathbb{F}_p^{n} \to \mathbb{F}_p^{n}$ by
G(x) = v + Mx. Define the k-fold composition of G by itself by $G^{1}(x) = G(x)$
and $G^{k+1} = G (G^{k}(x))$ Determine all pairs p,n for which there exists a vector v and a matrix M such that the $p^n$ vectors of $G^{k}(0), k=1,...,p^{n}$ are distinct.
 
Mathematics news on Phys.org
I've solved it but it will take a bit to type the justification for some of the points but contained in the spoiler are general tips for how to think about the problem
This is a pretty interesting question.
Basically it is asking, for which matrix group $GL_{n+1}(F_p)$, can you generate all the elements in $(F_p)^n$ by iterating a matrix (raising it to the power), and get all the elements in $(F_p)^n$.

Basically for which $GL_{n+1}(F_p)$ does there exist an element of order $p^{n}$.
 
Still writing it, hit sumbit by mistake, bear with me.
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Thread 'Imaginary Pythagorus'
I posted this in the Lame Math thread, but it's got me thinking. Is there any validity to this? Or is it really just a mathematical trick? Naively, I see that i2 + plus 12 does equal zero2. But does this have a meaning? I know one can treat the imaginary number line as just another axis like the reals, but does that mean this does represent a triangle in the complex plane with a hypotenuse of length zero? Ibix offered a rendering of the diagram using what I assume is matrix* notation...

Similar threads

Back
Top