I am investigating the existence of an orthogonal matrix satisfying specific conditions. Let $\mathbf{v}_1, \dots, \mathbf{v}_M \in \mathbb{R}^{2n}$ be real vectors with unit norm, where $M \leq 2n$. Additionally, let $\{\mathbf{e}_i\}^{2n}_{i=1}$ denote the canonical basis vectors of $\mathbb{R}^{2n}$. Consider the matrix $\Lambda$ defined as $$ \Lambda := \bigoplus_{i = 1}^{n} \begin{pmatrix} 0 & 1 \\ -1 & 0 \end{pmatrix}$$
I am seeking to establish the existence of an orthogonal matrix $O \in \mathrm{O}(2n)$ satisfying the following conditions:
- $O\Lambda O^T = \Lambda$,
- $\mathbf{e}_i^T O \mathbf{v}_j = 0$ for all $i \in \{2M+1, \dots, 2n \}$ and $j \in [M]$.
Here, $\mathrm{O}(2n)$ denotes the group of orthogonal matrices of size $2n \times 2n $.
I would appreciate any insights, proofs, or references that can shed light on the existence of such matrices. Note that, without requirement 1., the lemma would be a simple consequence of Gram-Schmidt. Requirement 1. means that the matrix $O$, in addition to being orthogonal, must be symplectic. I was therefore wondering if there is a simple proof of this fact by importing theorems from symplectic linear algebra.
Thank you!
