Constructing $g$-othogonal and symplectic basis of the vector space given a Lagrangian subspace.

Question

Assume that for a vector space $V$ we are given a fixed symplectic form $\omega$ and a fixed inner product $g$. Given Lagrangian subspace $\Lambda$ such that $g$-orthogonal complement $\Lambda^\perp$ is also Lagrangian, construct a $g$-orthogonal and $\omega$-standard basis.

My construction: pick any orthogonal basis $(e_1,\dots , e_n)$ of $\Lambda$, and denote $\Lambda_i$ a subspace of $\Lambda$ spanned by $(e_1,\dots, e_{i-1},e_{i+1},\dots,e_n)$ (ie. span of all basis vectors except $e_i$). Then intersection of symplectic complement $\Lambda_i^{\omega}$ and $\Lambda^{\perp}$ is a one dimensional subspace, because $\Lambda_i^{\omega}$ is $n+1$ dim and contains $\Lambda$. Then by non-degeneracy there exists a vector $f_i\in \Lambda_i^{\omega}\cap \Lambda^{\perp}$ such that $\omega(e_i,f_i)=1$.

Note that $\omega(e_i,f_j)=0$ if $i \neq j$ and $\omega(f_i,f_j)=0$, hence $(e_1,\dots,e_n,f_1,\dots,f_n)$ gives a symplectic basis. Since $f_i \in \Lambda^{\perp}$ we have that $g(e_i,f_j)=0$, however I don't see how to prove that $g(f_i,f_j)=0$ for $i\neq j$. It seems to me that the choice of $f_i$ in this process is canonical so I don't see how I could change it to ensure orthogonality between $f_i$'s

Answer 1

Regarding your method: Notice that you constructed the basis $\{f_1, \dots, f_n\}$ of $\Lambda^{\perp}$ without ever using $g|_{\Lambda^{\perp}}$, hence you can freely change $g$ on $\Lambda^{\perp}$ to produce examples for which $\{f_1, \dots, f_n\}$ is not $g$-orthogonal. One could be tempted to apply a Gram-Schmidt process to $\{f_1, \dots, f_n\}$ to obtain a $g$-orthogonal basis, but this wouldn't give an overall $\omega$-symplectic basis.

Note that the process you described is not completely canonical, since you begin with an arbitrarily chosen $g$-orthogonal basis $\{e_1, \dots, e_n\}$ for $\Lambda$. The conclusion is thus that you should construct the bases $\{e_1, \dots, e_n\}$ and $\{f_1, \dots, f_n\}$ 'simultaneously'.

Regarding the original problem: There are undoubtly several ways to solve the problem, and you should certainly try and find one for yourself. However, for completeness, here is one possible solution.

Start with $g$-orthonormal bases $E' = \{e'_1, \dots, e'_n\}$ of $\Lambda$ and $F' = \{f'_1, \dots, f'_n\}$ of $\Lambda^{\perp}$, thereby obtaining a $g$-orthonormal basis for $V$. With respect to this last basis, $\omega$ reads as the block matrix $\left( \begin{array}{cc} 0 & M \\ -M^{T} & 0 \end{array} \right)$ where $M$ is an invertible $(n \times n)$-matrix. (In fact, $\omega$ takes this form regardless of whether $E'$ and $F'$ are orthonormal or orthogonal, and any invertible matrix $M$ can appear here.) The goal is to change $E'$ and $F'$ within $\Lambda$ and $\Lambda^{\perp}$ respectively to obtain a merely $g$-orthogonal basis for $V$ with respect to which $M$ becomes $Id$. Now, we only have the liberty to transform $E'$ into a $g$-orthogonal basis $E$ for $\Lambda$, which is possible by acting via an automorphism $P_{\Lambda} : \Lambda \to \Lambda$ which is read (with respect to $E'$) as the matrix $O_{\Lambda} D_{\Lambda}$ where $O_{\Lambda}$ is orthogonal and $D_{\Lambda}$ is invertible diagonal. Similarly, we are allowed to change $F'$ into a $g$-orthonormal basis $F$ of $\Lambda^{\perp}$ via a map $P_{\Lambda^{\perp}} = O_{\Lambda^{\perp}}D_{\Lambda^{\perp}}$. Under this process, $M$ is changed into $P_{\Lambda}^T M P_{\Lambda^{\perp}}$, which we want to equal $Id$. This means we need to find $O_{\Lambda}$, $D_{\Lambda}$, $O_{\Lambda^{\perp}}$ and $D_{\Lambda^{\perp}}$ such that $M = O_{\Lambda} D_{\Lambda}D_{\Lambda^{\perp}}^T O_{\Lambda^{\perp}}^T$. The fact that $M$ can be expressed in this form follows from polar decomposition and spectral theorem.