Jacobi method for complex Hermitian matrices explained

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{align} (R _{pq} ) _{m,n} & = \delta _{m,n} & m,n \ne p,q, \\[ 10 pt] (R _{pq} ) _{p,p} & = (+1)/(\sqrt{ 2) } e^, \\[10pt](R_)_ & = \frac e^, \\[10pt](R_)_ & = \frac e^, \\[10pt](R_)_ & = \frac e^\end

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 \\[ 8 pt] (1)/(\sqrt{ 2) } (M_ e^ - M_ e^) & m = p \\[8pt]\frac (M_ e^ + M_ e^) & m = q\end \\[8pt](MR_^\dagger)_ & =\beginM_ & n \ne p,q \\\frac (M_ e^ - M_ e^) & n = p \\[8pt]\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 _{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} \\[ 6 pt] ⇒ R ^{\dagger ^{\dagger}} _{pq} & = R ^{-1 ^{\dagger}} _{pq} = R ^{-1 ^{-1}} _{pq} = R _{pq} . \end{align} T of a Hermitian matrix H is also a Hermitian matrix similar to H:\begin{align} T & \equiv R _{pq} H R ^{\dagger} _{pq} , & & \\[ 6 pt] 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} & = & & (H _{p,p} + H _{q,q})/(2) & - Re \{H _{p,q} e ^{-2i\theta} \}, \\[ 8 pt] T _{p,q} & = & & (H _{p,p} - H _{q,q})/(2) & + i Im \{H _{p,q} e ^{-2i\theta} \}, \\[ 8 pt] T _{q,p} & = & & (H _{p,p} - H _{q,q})/(2) & - i Im \{H _{p,q} e ^{-2i\theta} \}, \\[ 8 pt] T _{q,q} & = & & (H _{p,p} + H _{q,q})/(2) & + 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),with} \\[ 8 pt] \theta _{1} & \equiv (2 \phi _{1} - \pi)/(4) and \theta _{2} \equiv (\phi _{2})/(2) , \end{align}

where the phase terms,

\phi _{1} and \phi _{2} are given by:\begin{align} \tan \phi _{1} & = (Im \{H _{p,q} \)/() }, \\[8pt]\tan \phi_2 & = \frac.\end

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, \\[ 8 pt] - i e ^{- i\theta _{1}} \sin{\theta _{2}} & m = p and n = p, \\[ 8 pt] - e ^{+ i\theta _{1}} \cos{\theta _{2}} & m = p and n = q, \\[ 8 pt] e ^{- i\theta _{1}} \cos{\theta _{2}} & m = q and n = p, \\[ 8 pt] + i e ^{+ i\theta _{1}} \sin{\theta _{2}} & m = q and n = q. \end{cases} \end{align}

References

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