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 .

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 (R_)_ & = \delta_ & \qquad m,n \ne p,q, \\$0pt(R_)_ & = \frac e^, \\0pt(R_)_ & = \frac e^, \\0pt(R_)_ & = \frac e^, \\0pt(R_)_ & = \frac e^
\end

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
(R_ M)_ & =
\begin
M_ & m \ne p,q \\ pt\frac (M_ e^ - M_ e^) & m = p \\ pt\frac (M_ e^ + M_ e^) & m = q
\end \\ pt(MR_^\dagger)_ & =
\begin
M_ & n \ne p,q \\
\frac (M_ e^ - M_ e^) & n = p \\ pt\frac (M_ e^ + M_ e^) & n = q
\end
\end

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

:$H^\dagger = H \ \Leftrightarrow\ H_ = H^_$

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

:
\begin
R^\dagger_ & = R^_ \\ pt\Rightarrow\ R^_ & = R^_ = R^_ = R_.
\end

Hence, the complex equivalent Givens transformation $T$ of a Hermitian matrix ''H'' is also a Hermitian matrix similar to ''H'':

:
\begin
T & \equiv R_ H R^\dagger_, & & \\ ptT^\dagger & = (R_ H R^\dagger_)^\dagger = R^_ H^\dagger R^\dagger_ = R_ H R^\dagger_ = T
\end

The elements of ''T'' can be calculated by the relations above. The important elements for the Jacobi iteration are the following four:

:
\begin
T_ & = & & \frac & - \ \ \ \mathrm\, \\ ptT_ & = & & \frac & + \ i \ \mathrm\, \\ ptT_ & = & & \frac & - \ i \ \mathrm\, \\ ptT_ & = & & \frac & + \ \ \ \mathrm\.
\end

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
R^J_ & \equiv R_(\theta_2)\, R_(\theta_1),\text \\ pt\theta_1 & \equiv \frac \text \theta_2 \equiv \frac,
\end

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

:
\begin
\tan \phi_1 & = \frac, \\ pt\tan \phi_2 & = \frac.
\end

Finally, it is important to note that the product of two complex rotation matrices for given angles ''θ''₁ and ''θ''₂ cannot be transformed into a single complex unitary rotation matrix ''R''_''pq''(''θ''). The product of two complex rotation matrices are given by:

:
\begin
\left R_(\theta_2)\, R_(\theta_1) \right =
\begin
\ \ \ \ \delta_ & m,n \ne p,q, \\ pt-i e^\, \sin & m = p \text n = p, \\ pt- e^\, \cos & m = p \text n = q, \\ pt\ \ \ \ e^\, \cos & m = q \text n = p, \\ pt+i e^\, \sin & m = q \text n = q.
\end
\end

References

* .

{{Numerical linear algebra

Numerical linear algebra