Element-wise Formula for RoPE Rotation

A positional encoding method can be described in two ways. In one view, a d-dimensional vector is treated as d/2 complex numbers, and each complex number is rotated by an angle. In another view, the same d-dimensional vector is treated as d/2 pairs of real numbers, and each pair is rotated as a 2D vector by the same corresponding angle. What is the core principle that makes these two views equivalent?

Equivalence of 2D Vector Rotation Representations

Verifying Equivalence of 2D Rotational Methods

In a specific positional encoding method, pairs of embedding dimensions are rotated using a 2x2 matrix defined as: $R_{t\theta_k} = \begin{bmatrix} \cos t\theta_k & \sin t\theta_k -\sin t\theta_k & \cos t\theta_k \end{bmatrix}$ where t is the token's position and \theta_k is a frequency. Given a token at position t = 2 and a frequency \theta_k = \pi/4, which matrix correctly represents the rotation?

In a certain positional encoding method, the rotation of a pair of dimensions is achieved using the matrix $R = \begin{bmatrix} \cos(t\theta) & -\sin(t\theta) \sin(t\theta) & \cos(t\theta) \end{bmatrix}$, where t is the position and \theta is a frequency parameter.

Applying a Rotational Transformation Matrix