Abstract: It is a fundamental theorem in the theory of vibration that the inherent vibration of a linear vibration system has only non-negative real eigenvalues, and hence has real eigenvectors. This is usually referred to as the theory of the generalized eigenvalue problem. To the best knowledge of the author, most of the present textbooks on mechanical vibration theory give the proof of this problem based on the theory of the matrix factorization, which is often beyond the undergraduate students and makes it difficult for them to understand. On account of this situation, this paper presents an elementary proof for the generalized eigenvalue problem based on complex analysis with simple algebraic operations of matrix, without resorting to the theory of matrix factorization, but keeping the rigorousness and the integrity of the mathematical arguments.