For two n-by-n matrices $X$ and $Y$, we know that they are similar if the following is true for some invertible n-by-n matrix $M$:
$X = M^{-1}YM$.
Can anyone explain what the $M$ and $M^{-1}$ are doing? In other words, I'm finding plenty of information that contains this equation, however nobody seems to explain why we need to use both M and it's inverse. I know for a vector transformation we simply front-multiply by $M^{-1}$, but for matrices we must multiply on both sides. Why? Thanks!
This kind of operation is present in many different places in mathematics, and not only in linear algebra, when you have "to cross the mirror" ($M$ operation) for doing things more easily there (Y operation), and going back in the "real world" ($M^{-1}$ operation).
I will take a geometrical example: If you are willing to express the angle $\theta$ rotation around a center $C$ which is not the origin, you will do a shift (M operation = translation $\vec{MO}$) then rotate around the origin with this angle $\theta$ (Y operation) which is simple, then go back to the initial point $C$ ($M^{-1}$ operation = inverse translation $\vec{OM}$) (remark : a translation is not a linear operation).