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