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Greetings to all !

Let me first confess that this question was mentionned to me by Bernard Dacorogna, who doesn't sail on MO.

Let $A\in M_{2n}(k)$ be an alternate matrix. Say that $A$ is non-singular. It is well-known that there exists an $M\in GL_{2n}(k)$ such that $A=M^TJM$, where $$J=\begin{pmatrix} 0_n & I_n \\ - I_n & 0_n \end{pmatrix}.$$ Of course, $M$ is not unique. Every product $M'=QM$ with $Q\in Sp_n(k)$ ($Q$ symplectic) works as well.

Is it always possible to choose $M$ symmetric ? In this case, we have $A=MJM$, but an identity $A=RJR$ does not imply that $R$ be symmetric.

Equivalently,

Let $M\in GL_{2n}(k)$ be given. Is it true that $Sp_n(k)\cdot M$ meets $Sym_{2n}(k)$ non-trivially?

Notice that we must have $\det M=Pf(A)$. Therefore, if $k=\mathbb R$, the symmetric $M$ that we are looking for cannot always be positive definite.

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# Alternate and symmetric matrices

Greetings to all !

Let me first confess that this question was mentionned to me by Bernard Dacorogna, who doesn't sail on MO.

Let $A\in M_{2n}(k)$ be an alternate matrix. Say that $A$ is non-singular. It is well-known that there exists an $M\in GL_{2n}(k)$ such that $A=M^TJM$, where $$J=\begin{pmatrix} 0_n & I_n \\ - I_n & 0_n \end{pmatrix}.$$ Of course, $M$ is not unique. Every product $M'=QM$ with $Q\in Sp_n(k)$ ($Q$ symplectic) works as well.

Is it always possible to choose $M$ symmetric ? In this case, we have $A=MJM$, but an identity $A=RJR$ does not imply that $R$ be symmetric.

Equivalently,

Let $M\in GL_{2n}(k)$ be given. Is it true that $Sp_n(k)\cdot M$ meets $Sym_{2n}(k)$ non-trivially?

Notice that we must have $\det M=Pf(A)$. Therefore the symmetric $M$ that we are looking for cannot always be positive definite.