The complex unitary rotation matrices R pq 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 , R pq are defined as:
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{\displaystyle {\begin{aligned}(R_{pq})_{m,n}&=\delta _{m,n}&\qquad m,n\neq p,q,\\[10pt](R_{pq})_{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}}}
Each rotation matrix, R pq , will modify only the p th and q th rows or columns of a matrix M if it is applied from left or right, respectively:
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{\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}}}
A Hermitian matrix , H is defined by the conjugate transpose symmetry property:
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{\displaystyle H^{\dagger }=H\ \Leftrightarrow \ H_{i,j}=H_{j,i}^{*}}
By definition, the complex conjugate of a complex unitary rotation matrix, R is its inverse and also a complex unitary rotation matrix:
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{\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}}}
Hence, the complex equivalent Givens transformation
T
{\displaystyle T}
of a Hermitian matrix H is also a Hermitian matrix similar to H :
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{\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}}}
The elements of T can be calculated by the relations above. The important elements for the Jacobi iteration are the following four:
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{\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}}}
Each Jacobi iteration with R J pq generates a transformed matrix, T J , with T J p ,q = 0. The rotation matrix R J p ,q is defined as a product of two complex unitary rotation matrices.
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{\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}}}
where the phase terms,
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{\displaystyle \phi _{1}}
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{\displaystyle \phi _{2}}
are given by:
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{\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}}}
Finally, it is important to note that the product of two complex rotation matrices for given angles θ 1 and θ 2 cannot be transformed into a single complex unitary rotation matrix R pq (θ ). The product of two complex rotation matrices are given by:
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{\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}}}
Strang, G. (1993), Introduction to Linear Algebra , MA: Wellesley Cambridge Press .