Jacobi method for complex Hermitian matrices

Jacobi iteration method is generalized to complex Hermitian matrices.

Derivation

The complex unitary 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. $R_{pq}$ are defined as:

${\begin{array}{clrcl}(R_{pq})_{m,n}&=&&\delta _{m,n}&\qquad m,n\neq p,q,\\\\(R_{pq})_{p,p}&=&&{\frac {+1}{\sqrt {2}}}e^{-i\theta },&\\\\(R_{pq})_{q,p}&=&&{\frac {+1}{\sqrt {2}}}e^{-i\theta },&\\\\(R_{pq})_{p,q}&=&&{\frac {-1}{\sqrt {2}}}e^{+i\theta },&\\\\(R_{pq})_{q,q}&=&&{\frac {+1}{\sqrt {2}}}e^{+i\theta }&\end{array}}$

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:

${\begin{array}{clrcl}(R_{pq}M)_{m,n}&=&\left\{{\begin{array}{clrcl}M_{m,n}\qquad &m\neq &p,q\\{\frac {1}{\sqrt {2}}}(M_{p,n}e^{-i\theta }-M_{q,n}e^{+i\theta })\qquad &m=&p\\{\frac {1}{\sqrt {2}}}(M_{p,n}e^{-i\theta }+M_{q,n}e^{+i\theta })\qquad &m=&q\end{array}}\right\}\\(MR_{pq}^{\dagger })_{m,n}&=&\left\{{\begin{array}{clrcl}M_{m,n}\qquad &n\neq &p,q\\{\frac {1}{\sqrt {2}}}(M_{p,n}e^{+i\theta }-M_{q,n}e^{-i\theta })\qquad &n=&p\\{\frac {1}{\sqrt {2}}}(M_{p,n}e^{+i\theta }+M_{q,n}e^{+i\theta })\qquad &n=&q\end{array}}\right\}\end{array}}$

A Hermitian matrix, $M$ is defined by the conjugate transpose symmetry property:

$M^{\dagger }=M\ \Leftrightarrow \ M_{i,j}=M_{j,i}^{*}$

By definition, the complex conjugate of a complex unitary rotation matrix, $R$ is its inverse and also a complex unitary rotation matrix:

${\begin{array}{clrcl}&&R_{pq}^{\dagger }&=&R_{pq}^{-1}\\\\&\Rightarrow \ &R_{pq}^{\dagger ^{\dagger }}&=&R_{pq}^{-1^{\dagger }}=R_{pq}^{-1^{-1}}=R_{pq}.\end{array}}$

Hence, the complex equivalent Givens transformation $T$ of a Hermitian matrix $M$ is also a Hermitian matrix similar to $M$:

${\begin{array}{clrcl}T&\equiv &R_{pq}MR_{pq}^{\dagger },&&\\\\T^{\dagger }&=&(R_{pq}MR_{pq}^{\dagger })^{\dagger }&=&R_{pq}^{\dagger ^{\dagger }}M^{\dagger }R_{pq}^{\dagger }=R_{pq}MR_{pq}^{\dagger }=T\end{array}}$

The elements of $T$ can be calculated by the relations above. The important elements for the Jacobi iteration are the following four:

${\begin{array}{clrcl}T_{p,p}&=&&M_{p,q}&-\ \ \ Re\{M_{p,q}e^{-2i\theta }\},\\\\T_{p,q}&=&&{\frac {M_{p,p}-M_{q,q}}{2}}&+\ i\ Im\{M_{p,q}e^{-2i\theta }\},\\\\T_{q,p}&=&&{\frac {M_{p,p}+M_{q,q}}{2}}&-\ i\ Im\{M_{p,q}e^{-2i\theta }\},\\\\T_{q,q}&=&&M_{p,q}&+\ \ \ Re\{M_{p,q}e^{-2i\theta }\}.\end{array}}$

Each Jacobi iteration with $R_{pq}^{J}$ generates a transformed matrix, $T^{J}$, with $T_{p,q}^{J}=0$. The rotation matrix $R_{pq}^{J}$ is defined as a product of two complex unitary rotation matrices.

${\begin{array}{clrcl}R_{pq}^{J}&\equiv &R_{pq}(\theta _{2})\,R_{pq}(\theta _{1}),\ &{\text{ with}}&\\\\\theta _{1}&\equiv &{\frac {\pi -2i\phi _{1}}{4}}\ &{\text{ and}}&\\\\\theta _{2}&\equiv &{\frac {\phi _{2}}{2}},\ &&\end{array}}$

where the phase terms, $\phi _{1}$ and $\phi _{2}$ are given by:

${\begin{array}{clrcl}\tan {\phi _{1}}&=&{\frac {Im\{M_{p,q}\}}{Re\{M_{p,q}\}}},\\\\\tan {\phi _{2}}&=&{\frac {2M_{p,q}}{M_{p,p}-M_{q,q}}}.\end{array}}$

Finally, it is important to note that the product of two complex rotation matrices for given angles $\theta _{1}$ and $\theta _{2}$ cannot be transformed into a single complex unitary rotation matrix $R_{pq}(\theta )$. The product of two complex rotation matrices are given by:

${\begin{array}{clrcl}\left[R_{pq}(\theta _{2})\,R_{pq}(\theta _{1})\right]_{m,n}=\left\{{\begin{array}{clrcl}&\delta _{m,n}&m,n&\neq &p,q\\&-i\,e^{-i\theta _{1}}\,\sin {\theta _{2}}&m,n&=&p,p\\&-i\,e^{+i\theta _{1}}\,\cos {\theta _{2}}&m,n&=&p,q\\&e^{-i\theta _{1}}\,\cos {\theta _{2}}&m,n&=&q,p\\&+i\,e^{+i\theta _{1}}\,\sin {\theta _{2}}&m,n&=&q,q\end{array}}\right\}\end{array}}$