This paper deals with the joint diagonalization problem of a set of non-Hermitian matrices. It generalizes the first-order approximation used in the Fast Frobenius Diagonalization algorithm for Hermitian matrices to handle the non-Hermitian case. In the case of non-unitary diagonalization, the proposed Generalized (G)FFDiag algorithm outperforms the state-of-the-art algorithms and ensures fast convergence while reducing computational complexity. In the case of unitary diagonalization, we propose the Unitary (U)GFFDiag algorithm that follows the same framework as the GFFDiag algorithm and includes a final step that ensures the unitary constraint. Two variants are proposed depending on whether the unitary constraint is exact or approximated. The efficacy of GFFDiag and UGFFDiag is demonstrated by numerical experiments, including validation on real-world data, which indicate that these algorithms outperform existing algorithms in terms of computational efficiency and accuracy, particularly when dealing with a small number of large size matrices. The flexibility of these algorithms is highlighted by their applicability to both unitary and non-unitary diagonalization, as well as to rectangular and square matrices, thereby expanding their relevance across a wide range of applications.