The Rotary Position Embedding (RoPE) transformation can be represented in d-dimensional Euclidean space using the formula: $\mathrm{Ro}(\mathbf{x}, t\theta) = \begin{bmatrix} x_1 & x_2 & \dots & x_d \end{bmatrix} \begin{bmatrix} R_{t\theta_1} & & & \\ & R_{t\theta_2} & & \\ & & \ddots & \\ & & & R_{t\theta_{d/2}} \end{bmatrix}$ In this equation, a d-dimensional vector $\mathbf{x}$ is transformed by multiplying it with a block-diagonal matrix. This matrix consists of $d/2$ individual 2x2 rotation matrices, $R_{t\theta_k}$, along its diagonal. This structure effectively pairs up the components of $\mathbf{x}$ (e.g., $x_1$ with $x_2$, $x_3$ with $x_4$, etc.) and applies a separate 2D rotation to each pair.