Consider orthogonal matrices $Q_1, Q_2 \in \mathbb{R}^{d \times d}$, where both matrices are proper rotations of angle $\theta_1, \theta_2$ around different axes. Now, consider the symmetric matrix

$$ A = \frac{Q_1 Q_2 + Q_2^\top Q_1^\top}{2} $$

Question: is there a nontrivial lower bound for $\sigma_{\min}(A)$ in terms of $\theta_1, \theta_2$?

If $Q_1$ was the trivial rotation, we would have $\sigma_{\min}(A) \geq \cos(\theta_1)$. However, it is unclear to me what happens after composing the two rotations.


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