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