(**Edit** : see at the bottom of the question for an additional surprising possible hint.)

Using a computational software program, I found that the kernel of the following matrix is of dimension 2 when $n\geqslant 2$ but I haven't managed to prove it: \begin{equation} \text{for almost all } t_1>0,\quad \text{dim}\,\text{ker}\left(\mathbf{Q}_2\mathbf{Q}_1(t_1)-\mathbf{Q}_1(t_2)^{-1}\mathbf{Q}_2\right)\overbrace{=}^?\;2 \end{equation} where $t_2$ is chosen (assuming it exists) such that $$\text{det}\left(\mathbf{Q}_2\mathbf{Q}_1(t_1)-\mathbf{Q}_1(t_2)^{-1}\mathbf{Q}_2\right)=0$$ where

- $n\geqslant 2$
$\mathbf{Q}_2$ is the following matrix: \begin{equation} \mathbf{Q}_2=\begin{bmatrix} \mathbf{I}_n & \mathbf{0}_n \\ \mathbf{0}_n & \mathbf{P}^{-1}\begin{bmatrix}1 & && \\ & \ddots && \\ & & 1& \\ &&& -1 \end{bmatrix}\mathbf{P} \end{bmatrix}\in\mathbb{R}^{2n\times2n} \end{equation} where $\mathbf{P}\in\mathbb{R}^{n\times n}$ is any invertible matrix.

$\mathbf{Q}_1(t)$ is defined by:

\begin{equation} \forall t>0,\quad\mathbf{Q}_1(t)=\begin{bmatrix}\textbf{cos}(\boldsymbol \Omega t) & \boldsymbol \Omega^{-1}\,\textbf{sin}(\boldsymbol \Omega t) \\ -\boldsymbol \Omega\,\textbf{sin}(\boldsymbol \Omega t) & \textbf{cos}(\boldsymbol \Omega t)\end{bmatrix}\in\mathbb{R}^{2n\times2n} \end{equation} and: \begin{equation} \boldsymbol\Omega=\begin{bmatrix} \omega_1 & & 0\\ & \ddots & \\ 0 & & \omega_n \end{bmatrix}\in\mathbb{R}^{n\times n},\quad \forall i\in\lbrace 1,\dots, n\rbrace, \omega_i>0 \end{equation}

and the four blocks are diagonal, for example: \begin{equation} \mathbf{cos}(\boldsymbol\Omega t)=\begin{bmatrix} \cos(\omega_1t) & &0 \\ & \ddots & \\ 0 & & \cos(\omega_n t) \end{bmatrix}\in\mathbb{R}^{n\times n} \end{equation}

*Few properties of $\mathbf{Q}_1$ and $\mathbf{Q}_2$*:

Obviously, $\mathbf{Q}_2$ is invertible and $\mathbf{Q}_2=\mathbf{Q}_2^{-1}$.

Also, $\det(\mathbf{Q}_1)=1$ ($\omega_i>0$ and for proper $t>0$) and: \begin{equation} \mathbf{Q}_1(t)^{-1}=\begin{bmatrix}\textbf{cos}(\boldsymbol \Omega t) & -\boldsymbol \Omega^{-1}\,\textbf{sin}(\boldsymbol \Omega t) \\ \boldsymbol \Omega\,\textbf{sin}(\boldsymbol \Omega t) & \textbf{cos}(\boldsymbol \Omega t)\end{bmatrix} \end{equation}

Also, $\forall s,t,\ \mathbf{Q}_1(s+t)=\mathbf{Q}_1(s)\mathbf{Q}_1(t)=\mathbf{Q}_1(t)\mathbf{Q_1}(s)$.

(Note: I've already asked this question here but did not get any answer despite a bounty.)

**Edit** Additional possible hint:
I've found that for $n=2$ (maybe it's more general), the follow results stands, if $A=\mathbf{Q}_2\mathbf{Q}_1(t_1)-\mathbf{Q}_1(t_2)^{-1}\mathbf{Q}_2$ and $\phi(x)=\det(A-xI)$:
$$\phi'(0)^2=-\phi(0)\det(B)$$
where $B$ is the $A$ matrix if $\mathbf{P}=\mathbf{I}$... ! I have no clue where this comes from, maybe it's obvious for someone (maybe related to Lie algebras?).

This would conclude the proof as $\phi(0)=\det(A(t_1,t_2))=0$ for the appropriate $t_1,t_2$, so that $0$ would be a double eigenvalue of $A$.

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