Complex number λ and complex vector *z* are called an eigen pair of complex matrix *A*, if *Az = λz*. If matrix *A* of size NxN is Hermitian, it has *N* eigenvalues (not necessarily distinctive) and *N* corresponding eigenvectors which form an orthonormal basis (generally, eigenvectors are not orthogonal, and their number could be less than *N*).

For a Hermitian matrix *A*, there might be both the problem of finding all the eigenvalues and eigenvectors (the so-called matrix spectrum) and the problem of finding part of a spectrum. If not all the eigenvalues are required, we can use the bisection method to find the eigenvalues from a given interval (or having given indexes). After that, we can find the eigenvectors by using the inverse iteration method. If we only have to find a small part of the spectrum, we can increase the performance considerably in comparison with QL/QR algorithm. For more information see description of the similar algorithm for real symmetric matrices.

To find the eigenvalues (and their corresponding eigenvectors) from the given half-interval *(A, B]*, use the **HMatrixEVDR** subroutine. The **HMatrixEVDI** subroutine finds the eigen pairs having given numbers (the spectrum is considered as being sorted in ascending order).

It should be noted that the algorithm is effective only when finding a small part of the spectrum. If it is required to find all eigenvalues (or the majority of them), the QL/QR algorithm is more effective.

*This algorithm is transferred from the LAPACK 3.0*

*This article is licensed for personal use only.*

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