Summary: We discuss the accurate determination of matrix elements \(\langle f\mid \hat{h}_{\operatorname{W}}\mid i\rangle\) where neither \(|i\rangle\) nor \(|f\rangle\) is the vacuum state and \(\hat{h}_{\operatorname{W}}\) is some operator. Using solutions of the Generalized Eigenvalue Problem (GEVP) we construct estimators for matrix elements which converge rapidly as a function of the Euclidean time separations involved. \(|i\rangle\) and \(|f\rangle\) may be either the ground state in a given hadron channel or an excited state. Apart from a model calculation, the estimators are demonstrated to work well for the computation of the \(B^{\ast}B\pi\)-coupling in the quenched approximation. They are also compared to a standard ratio as well as to the “summed ratio method” of [L. Maiani et al., Nucl. Phys. B 293, 420 – 444 (1987); the authors, PoS(LATTICE 2010), Paper No. 303, 7 p. (2010), arXiv:1011.4393; S. Capitani et al, PoS(LATTICE 2010), Paper No. 147, 7 p. (2010); G.P. Lepage in T. DeGrand and D. Toussaint (eds.), From actions to answers, World Scientic, Singapore (1989); M. Lüscher, arxiv:1002.4232, 72 p. (2010)]. In the model, we also illustrate the ordinary use of the GEVP for energy levels.