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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).

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