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Let $A\in \Bbb F^{n\times m}$.

Let $\operatorname{neighborhood}(x)$ denote the elements surrounding $x$ ($x$ included). Let $a,b,c\in A$, $k\in \Bbb F$.

I've come across the following relation:

$$a\mathcal R_kb \iff a^2+\sum_ {c \in\operatorname {neighborhood(a)}}c^2=k.$$ Where $b$ is some element $\in \operatorname{neighborhood}(a)$.

I'm interested in knowing if this relation is an equivalence relation, it's easy to see that the relation is reflexive.

YoTengoUnLCD
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  • I am not sure this question makes much sense. What does it mean for something to be in a matrix? Why does it even matter that $A$ is a matrix? Why does $b$ not appear in the right hand side of the definition of the relation? – parsiad Sep 12 '15 at 18:22
  • Are you sure this is right? We don't have $b$ in $a^2+\sum_ {c \in\operatorname {neighborhood(a)}}c^2=k.$ – Ivo Terek Sep 12 '15 at 18:23
  • $b$ is supposed to be a number in the neighboorhood of $a$, this is how I've got it written down and it does look unclear, if anyone can help me clarify it, any help is welcome. – YoTengoUnLCD Sep 12 '15 at 18:24
  • If $a^{2} + \sum_{c \in neighborhood(a)} c^{2} \neq k$, then you do not have $aR_{k}a$ so it's clearly not an equivalence relation. – krirkrirk Sep 12 '15 at 18:24
  • You might want to use $$aR_{k}b \iff \sum_{c \in neighborhood(a)} c^{2} = \sum_{c \in neighborhood(b)} c^{2}$$ or something like that ? – krirkrirk Sep 12 '15 at 18:36

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