One way to prove that a matrix and its transpose are similar is to apply polynomial matrix theory.
I wonder if there is another way to do that. Thanks.
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3What do you mean by "polynomial matrix theory"? – carmichael561 Mar 13 '18 at 16:40
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@carmichael, I mean the theory of invariant polynomials – dmtri Mar 13 '18 at 17:12
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@Matteo, what if there is not such a canonical Jordan, for example when the field is not close? – dmtri Jun 16 '18 at 09:30
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The standard way to do it is to prove that $A$ and $A^t$ have the same Jordan canonical form. In turn, you can prove this showing that for all $\lambda$ and for all $n$
$(A-\lambda I)^n $ and $ (A^T-\lambda I)^n$ have the same rank.
Matteo Casarosa
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