I'd like to know whether the following statement is true or not.


Let $T\_1, T\_2\in \mathbb{C}^{n\times n}$ be upper triangular matrices. If there exists a nonsingular matrix $P$ such that $T\_1=PT\_2P^{-1}$, then there is a nonsingular uppper triangular matrix $T$ such that $T\_1=TT\_2T^{-1}$. 



a matrix similarity problem.
It is **false** for the following obvious reason. The diagonal elements of a triangular matrix are its eigenvalue. If they are pairwise distinct, the matrix is similar to its diagonal. 


Assume now that two upper triangular matrices have the same diagonal elements, pairwise distinct, but not in the same order. Then they are similar. Yet there does not exist an upper triangular $P$ such that $T\_2=PT\_1P^{-1}$, because conjugating by an upper triangular matrix does not change the diagonal elements.


This is related to the classification of flags.


