Euclidean Representation of RoPE Rotation

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

Consider a 4-dimensional real vector $\mathbf{x} = (1, 0, 0, 1)$. A transformation is applied that groups the vector's components into consecutive pairs, treats each pair as a complex number (a + ib), and rotates them independently. The first pair (components 1 and 2) is rotated by an angle of $\pi/2$ radians, and the second pair (components 3 and 4) is rotated by an angle of $\pi$ radians. What is the resulting vector after this transformation?

Determining an Unknown Rotation Angle

A transformation is applied to a d-dimensional real vector by grouping its components into d/2 consecutive pairs. Each pair is treated as representing a point in a 2D plane and is independently rotated by a specific, non-zero angle. True or False: This transformation will always change the Euclidean norm (magnitude) of the original d-dimensional vector.