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:

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

Each Jacobi iteration with $R_{pq}^{J}$ 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.

${\begin{aligned}R_{pq}^{J}&\equiv R_{pq}(\theta _{2})\,R_{pq}(\theta _{1}),&{\text{ with}}\\[8pt]\theta _{1}&\equiv {\frac {\pi -2i\phi _{1}}{4}}&{\text{ and}}\\[8pt]\theta _{2}&\equiv {\frac {\phi _{2}}{2}},&\end{aligned}}$

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

${\begin{aligned}\tan \phi _{1}&={\frac {\mathrm {Im} \{H_{p,q}\}}{\mathrm {Re} \{H_{p,q}\}}},\\[8pt]\tan \phi _{2}&={\frac {2H_{p,q}}{H_{p,p}-H_{q,q}}}.\end{aligned}}$

Finally, it is important to note that the product of two complex rotation matrices for given angles $\theta _{1}$ 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:

${\begin{aligned}\left[R_{pq}(\theta _{2})\,R_{pq}(\theta _{1})\right]_{m,n}={\begin{cases}\delta _{m,n}&m,n\neq p,q,\\[8pt]-ie^{-i\theta _{1}}\,\sin {\theta _{2}}&(m,n)=(p,p),\\[8pt]-ie^{+i\theta _{1}}\,\cos {\theta _{2}}&(m,n)=(p,q),\\[8pt]e^{-i\theta _{1}}\,\cos {\theta _{2}}&(m,n)=(q,p),\\[8pt]+ie^{+i\theta _{1}}\,\sin {\theta _{2}}&(m,n)=(q,q).\end{cases}}\end{aligned}}$

References

• Strang, G. (1993), Introduction to Linear Algebra, MA: Wellesley Cambridge Press.
• Akyuz, C. Deniz (1999), Resonant tunneling measurements of size-induced strain relaxation, ISBN 9780599327078. Ph.D Thesis, Department of Physics at Brown University.