There is the article on wikipedia https://en.wikipedia.org/wiki/Jacobi_method_for_complex_Hermitian_matrices about Jacobi eigenvalue algorithm for Hermitian matrices.
It says:
Similar to the Givens rotation matrices, $R_{pq}$ are defined as: $$ (R_{pq})_{m,n} = \delta_{m,n}, (m,n \ne p,q); \\ (R_{pq})_{p,p} = \frac{+1}{\sqrt{2}} e^{-i\theta}; \\ (R_{pq})_{q,p} = \frac{+1}{\sqrt{2}} e^{-i\theta}; \\ (R_{pq})_{p,q} = \frac{-1}{\sqrt{2}} e^{+i\theta}; \\ (R_{pq})_{q,q} = \frac{+1}{\sqrt{2}} e^{+i\theta} $$
But the usual rotation matrix is defined with cosinus and sinus like that: $$ \hat{\mathbf{R}} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} $$
This matrix rotates the vector $\mathbf{v} = \begin{pmatrix} x \\ y \end{pmatrix}$ on the angle $\theta$:
$$ R\mathbf{v} = \begin{pmatrix} \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{pmatrix} \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x\cos\theta - y\sin\theta \\ x\sin\theta + y\cos\theta \end{pmatrix}. $$
I don't understand why that complex rotation matrix $R_{pq}$ is defined with complex exponents and what exactly it rotates?
