In order to put forward a more convenient, effective, and practical method of diagonalizing Hamiltonian, the exact diagonalization of the Hamiltonian of the quantum boson model can be achieved through a general computation process in linear algebra, namely the N-ary quadratic form. Taking the Dicke model and the optomechanical hybrid system for example, the accuracy and effectiveness of this proposed method are verified through the linear algebra calculation process from the ordinary quadratic form to the standard quadratic one. Although this method is constrained by the second-order configuration for dealing with the Hamiltonian diagonalization, it is also believed to be able to provide another basic and effective method for studying the quantum phase transitions for some special systems.