Consider a 4-dimensional row vector $\mathbf{x} = \begin{bmatrix} 1 & 0 & 0 & 1 \end{bmatrix}$. This vector undergoes a rotational transformation by being post-multiplied by a 4x4 block-diagonal matrix. This matrix consists of two 2x2 rotation blocks along its diagonal. The first block rotates the first pair of vector components by an angle of $\pi/2$, and the second block rotates the second pair of components by an angle of $\pi$. Given that each 2x2 rotation block $R_{\alpha}$ for an angle $\alpha$ is defined as $\begin{bmatrix} \cos \alpha & \sin \alpha \\ -\sin \alpha & \cos \alpha \end{bmatrix}$, what is the resulting vector?

Consider the transformation of a d-dimensional vector by post-multiplication with a d-dimensional block-diagonal rotation matrix, as used in Rotary Positional Embeddings. This transformation will alter the Euclidean norm (magnitude) of the vector.

Independent Rotational Transformation in RoPE