# Generalized Symmetric-Definite Eigenvalue Problems

Generalized symmetric-definite eigenvalue problems are as follows: find the eigenvalues λ and the corresponding eigenvectors z that satisfy one of these equations:

`Az = λBz`, `ABz = λz`, or `BAz = λz`,

where A is an n-by-n symmetric or Hermitian matrix, and B is an n-by-n symmetric positive-definite or Hermitian positive-definite matrix.

In these problems, there exist n real eigenvectors corresponding to real eigenvalues (even for complex Hermitian matrices A and B).

Routines described in this topic allow you to reduce the above generalized problems to standard symmetric eigenvalue problem `Cy = λy`, which you can solve by calling LAPACK routines described earlier in this chapter (see Symmetric Eigenvalue Problems).

Different routines allow the matrices to be stored either conventionally or in packed storage. Prior to reduction, the positive-definite matrix B must first be factorized using either ?potrf or ?pptrf.

The reduction routine for the banded matrices A and B uses a split Cholesky factorization for which a specific routine ?pbstf is provided. This refinement halves the amount of work required to form matrix C.

Table “Computational Routines for Reducing Generalized Eigenproblems to Standard Problems” lists LAPACK routines (Fortran 77 interface) that can be used to solve generalized symmetric-definite eigenvalue problems. Respective routine names in Fortran 95 interface are without the first symbol (see Routine Naming Conventions).

Computational Routines for Reducing Generalized Eigenproblems to Standard Problems
Matrix type | Reduce to standard problems (full storage) | Reduce to standard problems (packed storage) | Reduce to standard problems (band matrices) | Factorize band matrix
--- | --- | --- | --- | ---
real symmetric matrices | ?sygst | ?spgst | ?sbgst | ?pbstf
complex Hermitian matrices | ?hegst | ?hpgst | ?hbgst | ?pbstf