$A \in \mathbb{C}^{n,n}$ is upper triangular and all it's elements on main diagonal are different. I have to prove that $A$ and $A^T$ are similar. I know that $A$ and $A^T$ have same characteristic polynomial but I don't know how to prove what I have to prove. Any hints?
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https://math.stackexchange.com/questions/94599/a-matrix-is-similar-to-its-transpose – jjjjjj Feb 17 '18 at 21:54
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1@C.Oliveira This is not a duplicate. The other question is much more general. – José Carlos Santos Feb 17 '18 at 21:57
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Let $a_1,\ldots,a_n$ be the entries of the main diagonal of $A$. Since they are distinct, $A$ is similar to the diagonal matrix whose entries of the main diagonal are $a_1,\ldots,a_n$. And the same thing applies to $A^T$. Therefore, $A$ and $A^T$ are similar.
José Carlos Santos
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