Jacobi method for complex Hermitian matrices

Introduction

Jacobi iteration method is generalized to complex Hermitian matrices. 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. $R_{pq}$ 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:

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

References

• Strang, G. (1993), Introduction to Linear Algebra, MA: Wellesley Cambridge Press.
• Akyuz, C. Deniz (1999), Resonant tunneling measurements of size-induced strain relaxation. Ph.D Thesis, Department of Physics at Brown University.