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From this post (and other similar ones) we know, in particular, that $$ \forall M\in{\rm GL}(n,\Bbb R),\ \exists P\in{\rm GL}(n,\Bbb R)\ \text{s.t. } PMP^{-1} = M^T\ . \tag{$*$} $$ My question is about the smoothness of the map $F:{\rm GL}(n,\Bbb R)\to{\rm GL}(n,\Bbb R)$ given by $M\mapsto P$, where $P$ satisfies $(*)$ above. Is $F$ smooth as a map between Lie groups?

From the answers to the above linked post, one possible way to construct $P$ (given an $M$) is using the Jordan Canonical Form. Then I suspect that the entries of $P$ may depend polynomially (or in general smoothly) on the entries of the corresponding matrix $M$. But I can't seem to find a reference for this or convince myself "more" rigorously that this is true. So, Is this true? TIA.

math-physicist
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  • Do you mean $F(M) = P$, where $PMP^{-1} = M^T$? – Matthew Leingang Jun 04 '24 at 21:03
  • @MatthewLeingang : oh yes! You're right. Very sorry. Can't believe I confused those. I've edited my post. – math-physicist Jun 04 '24 at 21:05
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    Is $P$ unique? I should know that, but I don't. If not, this isn't a function. But you might be able to use the implicit function theorem to show that, in the neighborhood of some $(M_0,P_0)$, $P$ depends smoothly on $M$. – Matthew Leingang Jun 04 '24 at 21:07
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    @MatthewLeingang Certainly not unique, see $n=1$. Moreover, if $P$ is a solution, then $\lambda P$ is another for all $\lambda\ne0$. – Sassatelli Giulio Jun 04 '24 at 21:13
  • @SassatelliGiulio right you are! – Matthew Leingang Jun 04 '24 at 21:24
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    Moreover, well, you are intersecting a vector subspace with $GL(n,\Bbb R)$, so in general we expect the solution set for $P_M$ to be a (non-empty) open subset of $\ker(X\mapsto XM-M^TX)$. – Sassatelli Giulio Jun 04 '24 at 21:35
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    As others have said, your $F$ is not well-defined. Basically you are considering $E={(M,P)\mid PMP^{-1}=M^T}$ and asking if the projection $\pi(M,P)=M$ has a smooth section. (Note the fiber $\pi^{-1}(M)$ is always a coset of ${\rm Stab}(M)$ so it varies with $M$.) – coiso Jun 04 '24 at 21:43
  • Thank you all for your comments. I think these comments answer my questions! – math-physicist Jun 04 '24 at 21:50

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