Jacobian ideal

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 Template:Harvtxt.

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:

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

Each rotation matrix, R_pq, will modify only the pth and qth rows or columns of a matrix M if it is applied from left or right, respectively:

$\begin{array}{rl}(R_{pq}M{)}_{m,n}& =\{\begin{array}{ll}{M}_{m,n}& m\ne p,q\\ \frac{1}{\sqrt{2}}(M_{p,n}{e}^{-i\theta }-M_{q,n}{e}^{+i\theta })& m=p\\ \frac{1}{\sqrt{2}}(M_{p,n}{e}^{-i\theta }+M_{q,n}{e}^{+i\theta })& m=q\end{array}\\ (M{R}_{pq}^{†}{)}_{m,n}& =\{\begin{array}{ll}{M}_{m,n}& n\ne p,q\\ \frac{1}{\sqrt{2}}(M_{m,p}{e}^{+i\theta }-M_{m,q}{e}^{-i\theta })& n=p\\ \frac{1}{\sqrt{2}}(M_{m,p}{e}^{+i\theta }+M_{m,q}{e}^{+i\theta })& n=q\end{array}\end{array}$

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

$H^{†}=H\iff H_{i,j}=H_{j,i}^{*}$

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

$\begin{array}{rl}{R}_{pq}^{†}& =R_{pq}^{-1}\\ \Rightarrow R_{pq}^{{†}^{†}}& =R_{pq}^{-1^{†}}=R_{pq}^{-1^{-1}}=R_{pq}.\end{array}$

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

$\begin{array}{rlrl}T& \equiv R_{pq}H{R}_{pq}^{†},& & \\ T^{†}& =(R_{pq}H{R}_{pq}^{†}{)}^{†}=R_{pq}^{{†}^{†}}{H}^{†}{R}_{pq}^{†}=R_{pq}H{R}_{pq}^{†}=T\end{array}$

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

$\begin{array}{clrcl}{T}_{p,p}& =& & H_{p,q}& -\mathrm{R}\mathrm{e}\{H_{p,q}{e}^{-2i\theta }\},\\ T_{p,q}& =& & \frac{H_{p,p}-H_{q,q}}{2}& +i\mathrm{I}\mathrm{m}\{H_{p,q}{e}^{-2i\theta }\},\\ T_{q,p}& =& & \frac{H_{p,p}+H_{q,q}}{2}& -i\mathrm{I}\mathrm{m}\{H_{p,q}{e}^{-2i\theta }\},\\ T_{q,q}& =& & H_{p,q}& +\mathrm{R}\mathrm{e}\{H_{p,q}{e}^{-2i\theta }\}.\end{array}$

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.

$\begin{array}{rl}{R}_{pq}^J& \equiv R_{pq}({\theta }_2)R_{pq}({\theta }_1),\text{ with}\\ {\theta }_1& \equiv \frac{\pi -2{\varphi }_1}{4}\text{ and }{\theta }_2\equiv \frac{{\varphi }_2}{2},\end{array}$

where the phase terms, ${\varphi }_1$ and ${\varphi }_2$ are given by:

$\begin{array}{rl}\tan {\varphi }_1& =\frac{\mathrm{I}\mathrm{m}\{H_{p,q}\}}{\mathrm{R}\mathrm{e}\{H_{p,q}\}},\\ \tan {\varphi }_2& =\frac{2H_{p,q}}{H_{p,p}-H_{q,q}}.\end{array}$

Finally, it is important to note that the product of two complex rotation matrices for given angles θ₁ and θ₂ cannot be transformed into a single complex unitary rotation matrix R_pq(θ). The product of two complex rotation matrices are given by:

$\begin{array}{r}{[R_{pq}({\theta }_2)R_{pq}({\theta }_1)]}_{m,n}=\{\begin{array}{ll}{\delta }_{m,n}& m,n\ne p,q,\\ -i{e}^{-i{\theta }_1}\sin {\theta }_2& m=p\text{ and }n=p,\\ -i{e}^{+i{\theta }_1}\cos {\theta }_2& m=p\text{ and }n=q,\\ e^{-i{\theta }_1}\cos {\theta }_2& m=q\text{ and }n=p,\\ +i{e}^{+i{\theta }_1}\sin {\theta }_2& m=q\text{ and }n=q.\end{array}\end{array}$

References

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