Intro:
Recently, in a Deep Learning class, I was asked to prove that circulant matrices are commutative under matrix multiplication (you may see where my problem comes from with this information). There was one part of my original approach to this proof that I could not figure out. I already submitted the assignment with a different approach, but I was very curious about what this proof would look like.
Problem Statement:
Let $g(a, b) = (a-b)\mod n$ where $n \in \mathbb N$. For any two integers $i, k$ such that $0 \leq i, k \leq n-1$. We define the set $S_{ik}$ of tuples as $S_{ik} = \{(g[i, j], g[j, k]): j \in \mathbb Z \text{ and } 0 \leq j \leq n-1\}.$ Prove that if $(x, y) \in S_{ik}$, then $(y, x) \in S_{ik}$.
Example:
Say $i = 2, k = 2$, and $n = 5$. Then, our set looks like $$ S_{22} = \{(2, 3), (1, 4), (0, 0), (4, 1), (3, 2)\} $$ We clear reflections of the tuples with $(2, 3),(3, 2)$ and $(1, 4), (4, 1)$. I've tried a fair number of examples and they all turn out with this property.
If anyone has any ideas on how one would prove this, or a good counter-example, it would definitely satisfy my curiosity here.
