Jacobi method for complex Hermitian matrices
Encyclopedia
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 .
of complex Hermitian matrices in order to find a numerical estimation of their eigenvectors and eigenvalues simultaneously.
Similar to the Givens rotation matrices, Rpq are defined as:
Each rotation matrix, Rpq, will modify only the pth and qth 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
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 RJpq generates a transformed matrix, TJ, with TJp,q = 0. The rotation matrix RJp,q is defined as a product of two complex unitary rotation matrices.
where the phase terms,
and 
are given by:
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 Rpq(θ). The product of two complex rotation matrices are given by:
Jacobi eigenvalue algorithm
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix...
. The Jacobi iteration method
Jacobi eigenvalue algorithm
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix...
is also explained in "Introduction to Linear Algebra" by .
Derivation
The complex unitary rotation matrices Rpq can be used for Jacobi iterationJacobi eigenvalue algorithm
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix...
of complex Hermitian matrices in order to find a numerical estimation of their eigenvectors and eigenvalues simultaneously.
Similar to the Givens rotation matrices, Rpq are defined as:
Each rotation matrix, Rpq, will modify only the pth and qth 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
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
Jacobi eigenvalue algorithm
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix...
are the following four:
Each Jacobi iteration
Jacobi eigenvalue algorithm
In numerical linear algebra, the Jacobi eigenvalue algorithm is an iterative method for the calculation of the eigenvalues and eigenvectors of a real symmetric matrix...
with RJpq generates a transformed matrix, TJ, with TJp,q = 0. The rotation matrix RJp,q is defined as a product of two complex unitary rotation matrices.
where the phase terms,
and are given by: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 Rpq(θ). The product of two complex rotation matrices are given by: