Jacobi method for complex Hermitian matrices

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:

:$H^\backslash dagger\; =\; H\; \backslash \; \backslash Leftrightarrow\backslash \; H\_\{i,j\}\; =\; H^\{*\}\_\{j,i\}$

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, $\backslash phi\_1$ and $\backslash phi\_2$ 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}

• {{Citation | last=Strang | first=G. | authorlink=Gilbert Strang | title=Introduction to Linear Algebra | publisher=Wellesley Cambridge Press | location = MA | year=1993}}.

{{Numerical linear algebra}}

Category:Numerical linear algebra