This paper is devoted to the study on the spectrum of Hermitizable tridiagonal matrices. As an illustration of the application of the author’s recent results on Hermitizable matrices, an explicit criterion for discrete spectrum of the matrices is presented, with a slight and technical restriction. The problem is well known, but from the author’s knowledge, it has been largely opened for quite a long time. It is important in various application, in quantum mechanics for instance. The main tool to solve the problem is the isospectral technique developed a few years ago. Two alternative constructions of the isospectral operator are presented; they are helpful in theoretical analysis and in numerical computations, respectively. Some illustrated examples are included.