[en] Modified braid equations satisfied by generalized R matrices (for a given set of group relations obeyed by the elements of T matrices) are constructed for q-deformed quantum groups GL_q(N), SO_q(N), and Sp_q(N) with arbitrary values of N. The Baxterization of R matrices, treated as an aspect complementary to the modification of the braid equation, is obtained for all these cases in particularly elegant forms. A new class of braid matrices is discovered for the quantum groups SO_q(N) and Sp_q(N). The R matrices of this class, while being distinct from the restrictions of the universal R matrix to the corresponding vector representations, satisfy the standard braid equation. The modified braid equation and the Baxterization are obtained for this new class of R matrices. Diagonalization of the generalized R matrices is studied. The diagonalizers are obtained explicitly for some lower dimensional cases in a convenient way, giving directly the eigenvalues of the corresponding R matrices. Applications of such diagonalization are then studied in the context of associated covariantly quantized noncommutative spaces