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:

$\begin{align} (R_{pq})_{m,n} & = \delta_{m,n} & \qquad m,n \ne p,q, \\[10pt] (R_{pq})_{p,p} & = \frac{+1}{\sqrt{2}} e^{-i\theta}, \\[10pt] (R_{pq})_{q,p} & = \frac{+1}{\sqrt{2}} e^{-i\theta}, \\[10pt] (R_{pq})_{p,q} & = \frac{-1}{\sqrt{2}} e^{+i\theta}, \\[10pt] (R_{pq})_{q,q} & = \frac{+1}{\sqrt{2}} e^{+i\theta} \end{align}$

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:

$\begin{align} (R_{pq} M)_{m,n} & = \begin{cases} M_{m,n} & m \ne p,q \\[8pt] \frac{1}{\sqrt{2}} (M_{p,n} e^{-i\theta} - M_{q,n} e^{+i\theta}) & m = p \\[8pt] \frac{1}{\sqrt{2}} (M_{p,n} e^{-i\theta} + M_{q,n} e^{+i\theta}) & m = q \end{cases} \\[8pt] (MR_{pq}^\dagger)_{m,n} & = \begin{cases} M_{m,n} & n \ne p,q \\ \frac{1}{\sqrt{2}} (M_{m,p} e^{+i\theta} - M_{m,q} e^{-i\theta}) & n = p \\[8pt] \frac{1}{\sqrt{2}} (M_{m,p} e^{+i\theta} + M_{m,q} e^{-i\theta}) & n = q \end{cases} \end{align}$

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:

$\begin{align} R^\dagger_{pq} & = R^{-1}_{pq} \\[6pt] \Rightarrow\ R^{\dagger^\dagger}_{pq} & = R^{-1^\dagger}_{pq} = R^{-1^{-1}}_{pq} = R_{pq}. \end{align}$

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

$\begin{align} T & \equiv R_{pq} H R^\dagger_{pq}, & & \\[6pt] T^\dagger & = (R_{pq} H R^\dagger_{pq})^\dagger = R^{\dagger^\dagger}_{pq} H^\dagger R^\dagger_{pq} = R_{pq} H R^\dagger_{pq} = T \end{align}$

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

$\begin{array}{clrcl} T_{p,p} & = & & \frac{H_{p,p} + H_{q,q}}{2} & - \ \ \ \mathrm{Re}\{H_{p,q} e^{-2i\theta}\}, \\[8pt] T_{p,q} & = & & \frac{H_{p,p} - H_{q,q}}{2} & + \ i \ \mathrm{Im}\{H_{p,q} e^{-2i\theta}\}, \\[8pt] T_{q,p} & = & & \frac{H_{p,p} - H_{q,q}}{2} & - \ i \ \mathrm{Im}\{H_{p,q} e^{-2i\theta}\}, \\[8pt] T_{q,q} & = & & \frac{H_{p,p} + H_{q,q}}{2} & + \ \ \ \mathrm{Re}\{H_{p,q} e^{-2i\theta}\}. \end{array}$

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.

$\begin{align} R^J_{pq} & \equiv R_{pq}(\theta_2)\, R_{pq}(\theta_1),\text{ with} \\[8pt] \theta_1 & \equiv \frac{2\phi_1 - \pi}{4} \text{ and } \theta_2 \equiv \frac{\phi_2}{2}, \end{align}$

where the phase terms, $\phi_1$ and $\phi_2$ are given by:

$\begin{align} \tan \phi_1 & = \frac{\mathrm{Im}\{H_{p,q}\}}{\mathrm{Re}\{H_{p,q}\}}, \\[8pt] \tan \phi_2 & = \frac{2 |H_{p,q}|}{H_{p,p} - H_{q,q}}. \end{align}$

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:

$\begin{align} \left[ R_{pq}(\theta_2)\, R_{pq}(\theta_1) \right]_{m,n} = \begin{cases} \ \ \ \ \delta_{m,n} & m,n \ne p,q, \\[8pt] -i e^{-i\theta_1}\, \sin{\theta_2} & m = p \text{ and } n = p, \\[8pt] - e^{+i\theta_1}\, \cos{\theta_2} & m = p \text{ and } n = q, \\[8pt] \ \ \ \ e^{-i\theta_1}\, \cos{\theta_2} & m = q \text{ and } n = p, \\[8pt] +i e^{+i\theta_1}\, \sin{\theta_2} & m = q \text{ and } n = q. \end{cases} \end{align}$

References

• Strang, G. (1993), Introduction to Linear Algebra, MA: Wellesley Cambridge Press.