Before and after a transformation apply another transformation and its inverse?

Question

So I have something like $y(x) = x\cdot5$, so whatever $x$ is, it's being scaled up by $5$.

But I could center that scaling to a different point reference other than $0$. For example $10$. So for that I end up with $y(x) = (x-10)\cdot 5+10$. I subtract $10$, making all $x$ values relative to $10$, I scale like before by multiplying by $5$, and then put things back by adding $10$. So it will look like instead of values are scaling from $0$, they are scaling from $10$.

Now, alternatively I can do this with multiplication instead of addition, so for example instead of $y(x) = \operatorname{floor}(x)$, I could increase the floor "step" size by $3$, doing $y(x) = \operatorname{floor}\left(\dfrac{x}{3}\right)\cdot3$.

And I find this kind of Give&Take or "compensation" routine to be applicable and useful with other operations.

I was just wondering if there was a proper name for this type of procedure, where you apply an operation and its inverese before and after a another operation.

Thank you!

Answer 1

This is called conjugation - specifically, the operation "before doing $A$, do $T$, and then after doing $A$, do $T^{-1}$" (more simply, "$A\mapsto T^{-1}AT$") is called conjugation by $T$ (or conjugation by $T^{-1}$, depending on the convention!).

Answer 2

Do you mean conjugation? In group theory, 'conjugating an element $a$ by $g$' means the element $gag^{-1}$.

See Conjugacy class.