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 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:

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