Jacobi method for complex Hermitian matrices

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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 Strang (1993).

Derivation[edit]

The complex unitary rotation matrices Rpq 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, 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:

References[edit]

  • Strang, G. (1993), Introduction to Linear Algebra, MA: Wellesley Cambridge Press .