The generalized eigenvalue problems are defined as , or , or , 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 1 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: Reduction of generalized symmetric definite eigenproblems to standard problems