The complex unitary rotation matrices Rpq can be used for Jacobi iteration of complex Hermitian matrices in order to find a numerical estimation of their eigenvectors and eigenvalues simultaneously.
Similar to the Givens rotation matrices, Rpq are defined as:
_{p,p}&={\frac {+1}{\sqrt {2}}}e^{-i\theta },\\[10pt](R_{pq})_{q,p}&={\frac {+1}{\sqrt {2}}}e^{-i\theta },\\[10pt](R_{pq})_{p,q}&={\frac {-1}{\sqrt {2}}}e^{+i\theta },\\[10pt](R_{pq})_{q,q}&={\frac {+1}{\sqrt {2}}}e^{+i\theta }\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/86a591846574a1d551c31bdac3735916893b3d0b)
Each rotation matrix, Rpq, will modify only the pth and qth rows or columns of a matrix M if it is applied from left or right, respectively:
![{\displaystyle {\begin{aligned}(R_{pq}M)_{m,n}&={\begin{cases}M_{m,n}&m\neq p,q\\[8pt]{\frac {1}{\sqrt {2}}}(M_{p,n}e^{-i\theta }-M_{q,n}e^{+i\theta })&m=p\\[8pt]{\frac {1}{\sqrt {2}}}(M_{p,n}e^{-i\theta }+M_{q,n}e^{+i\theta })&m=q\end{cases}}\\[8pt](MR_{pq}^{\dagger })_{m,n}&={\begin{cases}M_{m,n}&n\neq p,q\\{\frac {1}{\sqrt {2}}}(M_{m,p}e^{+i\theta }-M_{m,q}e^{-i\theta })&n=p\\[8pt]{\frac {1}{\sqrt {2}}}(M_{m,p}e^{+i\theta }+M_{m,q}e^{-i\theta })&n=q\end{cases}}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/733b2846efea6856cc9bc602c24362cd644c8076)
A Hermitian matrix, H is defined by the conjugate transpose symmetry property:
By definition, the complex conjugate of a complex unitary rotation matrix, R is its inverse and also a complex unitary rotation matrix:
![{\displaystyle {\begin{aligned}R_{pq}^{\dagger }&=R_{pq}^{-1}\\[6pt]\Rightarrow \ R_{pq}^{\dagger ^{\dagger }}&=R_{pq}^{-1^{\dagger }}=R_{pq}^{-1^{-1}}=R_{pq}.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/485440e737cc147e3c0d68854052d1f248646dd2)
Hence, the complex equivalent Givens transformation 
of a Hermitian matrix H is also a Hermitian matrix similar to H:
![{\displaystyle {\begin{aligned}T&\equiv R_{pq}HR_{pq}^{\dagger },&&\\[6pt]T^{\dagger }&=(R_{pq}HR_{pq}^{\dagger })^{\dagger }=R_{pq}^{\dagger ^{\dagger }}H^{\dagger }R_{pq}^{\dagger }=R_{pq}HR_{pq}^{\dagger }=T\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/669a674d71c101fe75c08515ea412e1a5b4b98bb)
The elements of T can be calculated by the relations above. The important elements for the Jacobi iteration are the following four:
![{\displaystyle {\begin{array}{clrcl}T_{p,p}&=&&{\frac {H_{p,p}+H_{q,q}}{2}}&-\ \ \ \mathrm {Re} \{H_{p,q}e^{-2i\theta }\},\\[8pt]T_{p,q}&=&&{\frac {H_{p,p}-H_{q,q}}{2}}&+\ i\ \mathrm {Im} \{H_{p,q}e^{-2i\theta }\},\\[8pt]T_{q,p}&=&&{\frac {H_{p,p}-H_{q,q}}{2}}&-\ i\ \mathrm {Im} \{H_{p,q}e^{-2i\theta }\},\\[8pt]T_{q,q}&=&&{\frac {H_{p,p}+H_{q,q}}{2}}&+\ \ \ \mathrm {Re} \{H_{p,q}e^{-2i\theta }\}.\end{array}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/73bda328adf3c06b42c41ba58846a5d8728b9b5d)
Each Jacobi iteration with RJpq generates a transformed matrix, TJ, with TJp,q = 0. The rotation matrix RJp,q is defined as a product of two complex unitary rotation matrices.
![{\displaystyle {\begin{aligned}R_{pq}^{J}&\equiv R_{pq}(\theta _{2})\,R_{pq}(\theta _{1}),{\text{ with}}\\[8pt]\theta _{1}&\equiv {\frac {2\phi _{1}-\pi }{4}}{\text{ and }}\theta _{2}\equiv {\frac {\phi _{2}}{2}},\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/f05b1c8690a2710e3e457b576a90d8ba8e3ef053)
where the phase terms, 
and 
are given by:
![{\displaystyle {\begin{aligned}\tan \phi _{1}&={\frac {\mathrm {Im} \{H_{p,q}\}}{\mathrm {Re} \{H_{p,q}\}}},\\[8pt]\tan \phi _{2}&={\frac {2|H_{p,q}|}{H_{p,p}-H_{q,q}}}.\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/3aef36674bd617dc2d5beeff49e89ebc0864e41a)
Finally, it is important to note that the product of two complex rotation matrices for given angles θ1 and θ2 cannot be transformed into a single complex unitary rotation matrix Rpq(θ). The product of two complex rotation matrices are given by:
![{\displaystyle {\begin{aligned}\left[R_{pq}(\theta _{2})\,R_{pq}(\theta _{1})\right]_{m,n}={\begin{cases}\ \ \ \ \delta _{m,n}&m,n\neq p,q,\\[8pt]-ie^{-i\theta _{1}}\,\sin {\theta _{2}}&m=p{\text{ and }}n=p,\\[8pt]-e^{+i\theta _{1}}\,\cos {\theta _{2}}&m=p{\text{ and }}n=q,\\[8pt]\ \ \ \ e^{-i\theta _{1}}\,\cos {\theta _{2}}&m=q{\text{ and }}n=p,\\[8pt]+ie^{+i\theta _{1}}\,\sin {\theta _{2}}&m=q{\text{ and }}n=q.\end{cases}}\end{aligned}}}](//wikimedia.org/api/rest_v1/media/math/render/svg/d5dd6b4ab3d788f29b0fe4424af2f6796fc929ba)
_{p,p}&={\frac {+1}{\sqrt {2}}}e^{-i\theta },\\[10pt](R_{pq})_{q,p}&={\frac {+1}{\sqrt {2}}}e^{-i\theta },\\[10pt](R_{pq})_{p,q}&={\frac {-1}{\sqrt {2}}}e^{+i\theta },\\[10pt](R_{pq})_{q,q}&={\frac {+1}{\sqrt {2}}}e^{+i\theta }\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/86a591846574a1d551c31bdac3735916893b3d0b)
![{\displaystyle {\begin{aligned}(R_{pq}M)_{m,n}&={\begin{cases}M_{m,n}&m\neq p,q\\[8pt]{\frac {1}{\sqrt {2}}}(M_{p,n}e^{-i\theta }-M_{q,n}e^{+i\theta })&m=p\\[8pt]{\frac {1}{\sqrt {2}}}(M_{p,n}e^{-i\theta }+M_{q,n}e^{+i\theta })&m=q\end{cases}}\\[8pt](MR_{pq}^{\dagger })_{m,n}&={\begin{cases}M_{m,n}&n\neq p,q\\{\frac {1}{\sqrt {2}}}(M_{m,p}e^{+i\theta }-M_{m,q}e^{-i\theta })&n=p\\[8pt]{\frac {1}{\sqrt {2}}}(M_{m,p}e^{+i\theta }+M_{m,q}e^{-i\theta })&n=q\end{cases}}\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/733b2846efea6856cc9bc602c24362cd644c8076)
![{\displaystyle {\begin{aligned}R_{pq}^{\dagger }&=R_{pq}^{-1}\\[6pt]\Rightarrow \ R_{pq}^{\dagger ^{\dagger }}&=R_{pq}^{-1^{\dagger }}=R_{pq}^{-1^{-1}}=R_{pq}.\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/485440e737cc147e3c0d68854052d1f248646dd2)
![{\displaystyle {\begin{aligned}T&\equiv R_{pq}HR_{pq}^{\dagger },&&\\[6pt]T^{\dagger }&=(R_{pq}HR_{pq}^{\dagger })^{\dagger }=R_{pq}^{\dagger ^{\dagger }}H^{\dagger }R_{pq}^{\dagger }=R_{pq}HR_{pq}^{\dagger }=T\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/669a674d71c101fe75c08515ea412e1a5b4b98bb)
![{\displaystyle {\begin{array}{clrcl}T_{p,p}&=&&{\frac {H_{p,p}+H_{q,q}}{2}}&-\ \ \ \mathrm {Re} \{H_{p,q}e^{-2i\theta }\},\\[8pt]T_{p,q}&=&&{\frac {H_{p,p}-H_{q,q}}{2}}&+\ i\ \mathrm {Im} \{H_{p,q}e^{-2i\theta }\},\\[8pt]T_{q,p}&=&&{\frac {H_{p,p}-H_{q,q}}{2}}&-\ i\ \mathrm {Im} \{H_{p,q}e^{-2i\theta }\},\\[8pt]T_{q,q}&=&&{\frac {H_{p,p}+H_{q,q}}{2}}&+\ \ \ \mathrm {Re} \{H_{p,q}e^{-2i\theta }\}.\end{array}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/73bda328adf3c06b42c41ba58846a5d8728b9b5d)
![{\displaystyle {\begin{aligned}R_{pq}^{J}&\equiv R_{pq}(\theta _{2})\,R_{pq}(\theta _{1}),{\text{ with}}\\[8pt]\theta _{1}&\equiv {\frac {2\phi _{1}-\pi }{4}}{\text{ and }}\theta _{2}\equiv {\frac {\phi _{2}}{2}},\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f05b1c8690a2710e3e457b576a90d8ba8e3ef053)
![{\displaystyle {\begin{aligned}\tan \phi _{1}&={\frac {\mathrm {Im} \{H_{p,q}\}}{\mathrm {Re} \{H_{p,q}\}}},\\[8pt]\tan \phi _{2}&={\frac {2|H_{p,q}|}{H_{p,p}-H_{q,q}}}.\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/3aef36674bd617dc2d5beeff49e89ebc0864e41a)
![{\displaystyle {\begin{aligned}\left[R_{pq}(\theta _{2})\,R_{pq}(\theta _{1})\right]_{m,n}={\begin{cases}\ \ \ \ \delta _{m,n}&m,n\neq p,q,\\[8pt]-ie^{-i\theta _{1}}\,\sin {\theta _{2}}&m=p{\text{ and }}n=p,\\[8pt]-e^{+i\theta _{1}}\,\cos {\theta _{2}}&m=p{\text{ and }}n=q,\\[8pt]\ \ \ \ e^{-i\theta _{1}}\,\cos {\theta _{2}}&m=q{\text{ and }}n=p,\\[8pt]+ie^{+i\theta _{1}}\,\sin {\theta _{2}}&m=q{\text{ and }}n=q.\end{cases}}\end{aligned}}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/d5dd6b4ab3d788f29b0fe4424af2f6796fc929ba)