Do there exist "families" of step-wise canonical transformations, in the sense that:
$${\bf A = TCT}^{-1}, \\{\bf T = T_2}^n$$
So that $${\bf T_2}^k {\bf C} {\bf T_2}^{-k} $$ Also has some canonical structure $\forall k\in\{1,\cdots, n\}$ in some sense similar to $\bf C$.
I realize this question is extremely vague/fuzzy, and I am willing to narrow the question down if either of us can help me do that.
Edit own sandbox example:
Consider the matrices $${\bf T_2}=\left[\begin{array}{cc}1&1\\0&1\end{array}\right] , {\bf C} = \left[\begin{array}{cc}0&a\\0&b\end{array}\right]$$
Now we can verify :
$${\bf T_2}^k{\bf CT_2}^{-k} = \left[\begin{array}{cc}0&a+bk\\0&b\end{array}\right]$$
Basically mimicking a generic linear function (or affine, actually).
