Flow of D.E what is the idea behind conjugacy? I got some kinda flow issue, ya know?
well enough with the bad jokes let A be a 2x2 matrix, T a change of Coordinate matrix, and $B=T^{-1}AT$ the canonical matrix ascoiated with A. Show that the function $h=T^{-1}:  \mathbb R^{2} \to  \mathbb R^{2}$, $h(x)=T^{-1}X$ is a conjugacy between the flows of the systems $X^{'}$= $AX$ and $Y^{'}$= $BY$
Firstly Is there any chance someone could quantify this in terms of algebraic set theory for me? i seem to half comprehend flow of dynamical systems and half understand there counter part in group theory. they feel so similar yet i can't seem to put the 2 together. it feels like flow of this system is a Monomorphism as its a homomorphism and it is a one-one mapping.
Second question how do i prove the above is a conjugacy between the flows, the method we were shown in class was very odd and didn't involve using T and $T^{-1}$i have asked a somewhat similar question and someone was kind enough to show me how to do it using this kind of idea but with numbers. i don't actually understand why this concept works.
All that i really understand is that if the same number of eigenvalues with positive sign are in each matrix there should be a mapping of the flow that exists between the two of them.

Suppose $X' = AX$.  Let $Y = T^{-1} X T$.  Then $$ Y' = (T^{-1} X T)' = T^{-1} X' T = T^{-1} (AX) T = (T^{-1} A T) (T^{-1} X T) = B (T^{-1} X T) = BY$$
So the flows of $B$ are simply conjugates of the flows of $A$.  Is this what you wanted to know?
