Square root of Symplectic and Positive Definite Matrices in $M_{2n\times 2n}(\mathbb{R})$ Let $R$ be a symplectic matrix. Then the adjoint $R^*$ of $R$ is also a symplectic matrix. Then $RR^*$ is symplectic and positive definite. I want to know that $(RR^*)^{1/2}$ is also symplectic.

Write $R=AU$, $A$ is symmetric positive definite $U$ is orthogonal.

$R^*$ is symplectic, so $J=R^*JR=U^*(AJA)U$, ie $AJA=UJU^*$, thus, with $J’=UJU^*$ orthogonal, we have $V:= AJ=J’A^{-1}$.

The conclusion then stems from the following lemma: if $X=AU=WB$ is an invertible matrix, $U,W$ are orthogonal, $A,B$ are symmetric positive semidefinite, then $U=W$. Indeed, if the lemma holds, then $J=J’$ and $U$ is symplectic, therefore so is $A=RU^{-1}$.

The proof runs as follows: we then have $X=U(U^*AU)=WB$, where $U,W$ are orthogonal, $A’=U^*AU$ and $B$ are positive semi-definite, so $BA’^{-1}$ is orthogonal, ie $A’^{-1}B^2A’^{-1}=I$, so $B^2=A’^2$, so (symmetric positive definite) $B=A’$ so $U=W$.