# Jacobi method for complex Hermitian matrices

In mathematics, the Jacobi method for complex Hermitian matrices is a generalization of the Jacobi iteration method. The Jacobi iteration method is also explained in "Introduction to Linear Algebra" by Strang (1993).

## Derivation

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:
    ${\begin{aligned}(R_{pq})_{m,n}&=\delta _{m,n}&\qquad m,n\neq p,q,\\[10pt](R_{pq})_{p,p}&={\frac {+1}{\sqrt {2}}}e^{-i\theta },\\[10pt](R_{pq})_{q,p}&={\frac {+1}{\sqrt {2}}}e^{-i\theta },\\[10pt](R_{pq})_{p,q}&={\frac {-1}{\sqrt {2}}}e^{+i\theta },\\[10pt](R_{pq})_{q,q}&={\frac {+1}{\sqrt {2}}}e^{+i\theta }\end{aligned}}$
Each rotation matrix, R$_{pq}$, will modify only the pth and qth rows or columns of a matrix M if it is applied from left or right, respectively:
    ${\begin{aligned}(R_{pq}M)_{m,n}&={\begin{cases}M_{m,n}&m\neq p,q\\[8pt]{\frac {1}{\sqrt {2}}}(M_{p,n}e^{-i\theta }-M_{q,n}e^{+i\theta })&m=p\\[8pt]{\frac {1}{\sqrt {2}}}(M_{p,n}e^{-i\theta }+M_{q,n}e^{+i\theta })&m=q\end{cases}}\\[8pt](MR_{pq}^{\dagger })_{m,n}&={\begin{cases}M_{m,n}&n\neq p,q\\{\frac {1}{\sqrt {2}}}(M_{m,p}e^{+i\theta }-M_{m,q}e^{-i\theta })&n=p\\[8pt]{\frac {1}{\sqrt {2}}}(M_{m,p}e^{+i\theta }+M_{m,q}e^{-i\theta })&n=q\end{cases}}\end{aligned}}$
A Hermitian matrix, H is defined by the conjugate transpose symmetry property:
    $H^{\dagger }=H\ \Leftrightarrow \ H_{i,j}=H_{j,i}^{*}$
By definition, the complex conjugate of a complex unitary rotation matrix, R is its inverse and also a complex unitary rotation matrix:
    ${\begin{aligned}R_{pq}^{\dagger }&=R_{pq}^{-1}\\[6pt]\Rightarrow \ R_{pq}^{\dagger ^{\dagger }}&=R_{pq}^{-1^{\dagger }}=R_{pq}^{-1^{-1}}=R_{pq}.\end{aligned}}$
Hence, the complex equivalent Givens transformation $T$ of a Hermitian matrix H is also a Hermitian matrix similar to H:
    ${\begin{aligned}T&\equiv R_{pq}HR_{pq}^{\dagger },&&\\[6pt]T^{\dagger }&=(R_{pq}HR_{pq}^{\dagger })^{\dagger }=R_{pq}^{\dagger ^{\dagger }}H^{\dagger }R_{pq}^{\dagger }=R_{pq}HR_{pq}^{\dagger }=T\end{aligned}}$
The elements of T can be calculated by the relations above. The important elements for the Jacobi iteration are the following four:
    ${\begin{array}{clrcl}T_{p,p}&=&&{\frac {H_{p,p}+H_{q,q}}{2}}&-\ \ \ \mathrm {Re} \{H_{p,q}e^{-2i\theta }\},\\[8pt]T_{p,q}&=&&{\frac {H_{p,p}-H_{q,q}}{2}}&+\ i\ \mathrm {Im} \{H_{p,q}e^{-2i\theta }\},\\[8pt]T_{q,p}&=&&{\frac {H_{p,p}-H_{q,q}}{2}}&-\ i\ \mathrm {Im} \{H_{p,q}e^{-2i\theta }\},\\[8pt]T_{q,q}&=&&{\frac {H_{p,p}+H_{q,q}}{2}}&+\ \ \ \mathrm {Re} \{H_{p,q}e^{-2i\theta }\}.\end{array}}$
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.
    ${\begin{aligned}R_{pq}^{J}&\equiv R_{pq}(\theta _{2})\,R_{pq}(\theta _{1}),{\text{ with}}\\[8pt]\theta _{1}&\equiv {\frac {2\phi _{1}-\pi }{4}}{\text{ and }}\theta _{2}\equiv {\frac {\phi _{2}}{2}},\end{aligned}}$
where the phase terms, $\phi _{1}$ and $\phi _{2}$ are given by:
    ${\begin{aligned}\tan \phi _{1}&={\frac {\mathrm {Im} \{H_{p,q}\}}{\mathrm {Re} \{H_{p,q}\}}},\\[8pt]\tan \phi _{2}&={\frac {2|H_{p,q}|}{H_{p,p}-H_{q,q}}}.\end{aligned}}$
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:
    ${\begin{aligned}\left[R_{pq}(\theta _{2})\,R_{pq}(\theta _{1})\right]_{m,n}={\begin{cases}\ \ \ \ \delta _{m,n}&m,n\neq p,q,\\[8pt]-ie^{-i\theta _{1}}\,\sin {\theta _{2}}&m=p{\text{ and }}n=p,\\[8pt]-e^{+i\theta _{1}}\,\cos {\theta _{2}}&m=p{\text{ and }}n=q,\\[8pt]\ \ \ \ e^{-i\theta _{1}}\,\cos {\theta _{2}}&m=q{\text{ and }}n=p,\\[8pt]+ie^{+i\theta _{1}}\,\sin {\theta _{2}}&m=q{\text{ and }}n=q.\end{cases}}\end{aligned}}$