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:
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:
A Hermitian matrix, "H" is defined by the conjugate transpose symmetry property:
By definition, the complex conjugate of a complex unitary rotation matrix, "R" is its inverse and also a complex unitary rotation matrix:
Hence, the complex equivalent Givens transformation formula_5 of a Hermitian matrix "H" is also a Hermitian matrix similar to "H":
The elements of "T" can be calculated by the relations above. The important elements for the Jacobi iteration are the following four:
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.
where the phase terms, formula_9 and formula_10 are given by:
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: