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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 $T_1=TT_2T^{-1}$.

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up vote 9 down vote accepted

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.

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It is also false for matrices with zero (or just constant) diagonal. – Petya Dec 16 '10 at 20:44

It is not true.

Let upper triangular matrix $T$ be also a differential, i.e. $T^2=0$. Then, acting by upper triangular conjugation one can get a matrix with at most one $1$ in each row and each column, all other elements equal to zero. Moreover, such a matrix (normal form) is unique in the upper triangular conjugacy class. But from the point of view of all conjugations two such a differentials are equivalent if and only if the dimensions of its homologies coincides.

The simplest example through differentials are matrices








They are not in the same upper triangular conjugacy class.

Remark: Such differentials and an action of the upper-triangular group naturally arise in a context of Morse theory. A strong Morse function and a generic Riemannian metric on a closed manifold generates Morse complex with an upper triangular Morse differential (upper triangular with respect to the ordering by critical value). A change of a metric reflects as an upper-triangular conjugation of Morse differential. The statement above (on the unicity of normal form in a case of field coefficients) was formulated and proved by Barannikov and attributed to J.Cerf (S. A. Barannikov, “The framed Morse complex and its invariants”, Adv. Soviet Math. 21. (1994), 93–115.).

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One more possible explanation of that phenomenon: upper triangular conjugation preserves the dimension of the intersection of the kernel of $T$ with a subspace spanned on first $l$-vectors. Thus, it is impossible to get (10)(00) from (00)(01) by an upper triangular conjuation. – Petya Dec 16 '10 at 20:21

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