Let formula_2 be a symmetric and formula_301 a symmetric and positive definite matrix. Write the generalized eigenvalue equation as formula_302 where we impose that formula_46 be normalized, i.e. formula_304. Now we use Cholesky decomposition to write the inverse of formula_301 as formula_306. Multiplying by formula_174 and letting formula_308, we get formula_309, which can be rewritten as formula_310 where formula_311. Manipulation now yields formula_312 where formula_313 is a matrix having as columns the generalized eigenvectors and formula_314 is a diagonal matrix of the generalized eigenvalues. Now premultiplication with formula_315 gives the final result: formula_316 and formula_317, but note that this is no longer an orthogonal diagonalization with respect to the inner product where formula_311. In fact, we diagonalized formula_2 with respect to the inner product induced by formula_301.