Then the generalized eigenvalue problem for is equivalent to the standard HEP for . Both share the same eigenvalues since

which also says that if is an eigenvector for the pair, is an eigenvector for the matrix , and on the other hand if is an eigenvector for , is an eigenvector for the pair. Common choices for are:

- , the unique positive definite square root of . In this case, . This choice is good enough for theoretical investigations.
- is the Cholesky factor; optionally with pivoting, i.e., is lower triangular with positive diagonal entries. This choice is preferred for numerical computations.
- Analogously is upper triangular with positive diagonal entries. It shares the same advantage of the second choice.

When , all three reduce to the usual definitions. It is rather easy to see that

With some extra work, we can relate to the usual angle function, e.g., for , as follows.