In this paper, the notions of equal convergence (ec), uniform equal convergence (u.ec), discrete convergence (dc), and uniform discrete convergence (u.dc), which were defined for the sequences of real-valued functions, are generalized with regard to any regular matrix $A=(a_{n,k})$, and their generalized form are studied. The classical and generalized versions of these convergence concepts are compared, and some inclusions are given. Through constructed examples, it is shown that inclusions between them are strict. Finally, as an application, a more general form of the famous Korovkin's theorem is presented.