This section is concerned with the solution of the generalized eigenvalue
problems , , and , where
*A* and *B* are real symmetric or complex Hermitian and *B* is positive definite.
Each of these problems can be reduced to a standard symmetric
eigenvalue problem, using a Cholesky factorization of *B* as either
or ( or in the Hermitian case).

With , we have

Hence the eigenvalues of are those of ,
where *C* is the symmetric matrix and .
In the complex case *C* is Hermitian with and .

Table 3.11 summarizes how each of the three types of problem
may be reduced to standard form
, and how the eigenvectors *z*
of the original problem may be recovered from the eigenvectors *y* of the
reduced problem. The table applies to real problems; for complex problems,
transposed matrices must be replaced by conjugate transposes.

**Table 3.11:** Reduction of generalized symmetric definite eigenproblems to standard
problems

Given *A* and a Cholesky factorization of *B*,
the routines PxyyGST overwrite *A*
with the matrix *C* of the corresponding standard problem
(see table 3.12).
This may then be solved by using the routines described in
subsection 3.3.4.
No special routines are needed
to recover the eigenvectors *z* of the generalized problem from
the eigenvectors *y* of the standard problem, because these
computations are simple applications of Level 2 or Level 3 BLAS.

**Table 3.12:** Computational routines for the generalized symmetric definite eigenproblem

Tue May 13 09:21:01 EDT 1997