Assume we have two matrices $A,B$ for which: $$A = TDT^{-1}\\B = T^{-1}DT$$ Can we derive anything about these matrices relation to each other?
I have tried experimenting a bit but I can't see anything special:
We can see that $$A^{k} = TD^kT^{-1}\\B^{k}=T^{-1}D^kT$$
$$AB = TDT^{-2}DT\\$$
However if we had an operation which flips all factors of matrix products without transposing them, then we can turn $A$ into $B$ and vice versa. But what kind of weird operation would that be?